Does it actually take infinite (observer) time for someone to fall into a black hole? If you were to watch your friend approach a black hole, I understand that you'd see their clock slow until they appear frozen and redshift within a few seconds. But if you were to detect the increasingly long wavelengths coming off of your friend, you would still see them frozen above the event horizon for infinite time. I want to know if this is due to actual time dilation and the friend is actually above the event horizon for infinite time or if this is because of some Doppler effect and the friend actually fell in but you're only observing the past events of the friend. Put another way, given any finite time, could you theoretically go fast enough to save your friend before they fall in? Or would you only realize that your friend fell in a long time ago?
 A: It doesn't. Strictly speaking, it takes undefined (or arbitrarily defined) time. For that matter, so does anything else happening at any remote point in a general curved space-time.
You see, in order to talk about "what time things take place at" or "how much time happens between two things", we need to first have a notion of "something happens at time $t$" (so we can say both "what happens at time $t_1$" and "what happens at time $t_2$" to talk about something "happening between times $t_1$ and $t_2$" so as to "take time $t = t_2 - t_1$".) that applies at every point within the space-time or, at least, at the two different locations we are interested in. That is, we need a definition of a simultaneity.
However there is no natural definition of a simultaneity in a general curved space-time! Whether it is a black hole or even just the regular curved space time of the Universe. Merely choosing an observer alone is not enough; there is way more degrees of freedom than that. Hence it is best to say that we can't really ask this question unless we specify that ahead of time, and then the answer depends entirely on what arbitrary process of simultaneity ascertainment we use.
But of course, there's clearly an obvious intended meaning behind this question which we'll get to, but to really address an deal with it, we first need a framework to ask the "proper" questions that fully fit within the relativity theory, then think about what questions within that that our colloquial intuition and understanding of this need asking. And the changes we need to make are fairly radical.
Instead of thinking about clocks and "universals", we have to think only about events and local regions (the maxim is "relativity requires us to talk in the language of events, not things" - but unfortunately, a lot of teaching does not catch up to that): each space-time point is what we call an "event", and a "local region" means a tiny bubble of space-time around that point.
At an event, information can either be received, or it can be transmitted, or both, i.e. passed from one event to a neighboring event. It cannot, however, just jump "instantly" from one point to another without passing through all the succeeding points, and that is in general part of why we cannot talk about such a "now" at least not in any observable sense. Moreover, even were such "instant" jumpings possible, there is no inherent necessity for them to occur in any particular order at all and thus to define a coherent "now" solely based on the geometry of the space-time in question alone.
And this then has great ramifications for what we perceive. The way we perceive the Universe is that information has to hop to us from other points in space-time following some such unbroken chain of linkages. For example, when we literally see things, the chain of linkages is that photons of light have to move from one point to another point to another until they finally reach our eyes. There is not any "jumping" from one to the other.
This is called locality. If such a set of "jumpings" is possible to reach from one event to our eye, then we can see what is happening there. If it isn't, we can't.
One more piece to mention is that this does not in any way stop observers themselves from having their own local notions of time: indeed, this is THE proper way to talk of "time" in relativity and it is no surprise it is called the "proper time" - the concept of time that is fit or suitable for them in a natural and non-arbitrary way. No observer can talk non-arbitrarily about times when things occur far-off, but they can talk about a succession of times happening within their own "self". And discussing their whole experience of the Universe is entirely about discussing what information receipts occur and at what times along that proper time line, and why they do or do not.
Of course, you may then ask, "well, if this is the case, why does all our everyday experience seem to tell us otherwise?" It's simple. What we see with our eyes happens as a result of them getting light signals. And light travels so fast that, between any two objects on Earth, the time for a round-trip light message is imperceptible and this creates an approximate notion of simultaneity - but it is again "local" when compared to the rest of the Universe, where Earth is pointlike (think in petameters, not kilometers, for example: Earth is $0.000\thinspace000\thinspace012\thinspace743\ \mathrm{Pm}$ in diameter. This is tiny.).
We can now go back to the black hole. The question that we are thinking of is really motivated by the answer to this question: "if someone falls into a black hole, what will I see?". And yes, the answer to this is that "you will see them slow down and effectively 'freeze' just above the horizon". And no, this fact - note it is about "what you see" - does not depend on any arbitrary choices about how to talk about things in the space-time.
But this does not mean they are "above it for an infinite time" "actually", because as said, we cannot talk about what "actually" is going on "out there" "now" because there is no notion of "now" that simultaneously applies both here and there that is not arbitrary and hence unequivocally and undebatably "how things really are", at least not within the framework of relativity theory. It also does not mean they don't fall into the black hole. In the language we just developed, "fall into the black hole" means "hop through a series of spacetime points that leads from the outside to the inside solely under the guidance of your four-momentum, with no other influences". And that is always possible and does happen. It's just that you can't assign, as someone else at a far away point, any "when" values to these in a non arbitrary manner. You can, of course, use what you see - which is, as I said above, NOT arbitrary, as one possible assignment, and then you will indeed get the answer "it takes infinite amount of time for them to fall in". But there is nothing sacrosanct about this and simply specifying. It is purely the result of taking an arbitrary choice to go by what you see, which is a hugely non-trivial choice as it fixes all the uncountably many degrees of freedom I just mentioned above.
Of course, you then may still ask, "is there a reason for why we see this?" And of course, there is. But to do it full justice then requires us to think a bit more closely about the notion of an event horizon in terms of the language we've just set up. An event horizon is a space-time surface (not just a spatial one), such that there are hopping sequences going from events outside to events inside, but not hopping sequences going from events outside to events inside. That's it; nothing more! It's that simple.
Hence, if someone carrying a light source suicidally drops into the black hole, "once" (i.e. by their proper time) they pass that horizon, any photons that light source is sending out, will not go out, and before that point, they can (and if they are aimed away from the hole, they will).
And this is where it gets somewhat profound. The trick is, the photons are leaving in just such a way that for those which are emitted just before crossing, they will arrive at a distant observer only at far more remote proper times compared to if they were emitted further away. There is no "time dilation" (NOTE: see added blurb), rather, there is just the way photons are moving, and an absence of "time" as an non-arbitrary global concept. And the reason for this is that it, in effect, is in part the result of a deep logical constraint regarding information. You see, since we defined everything in terms of the movement of information, depending on how you understand "information", you can say that a piece of information you receive is not just informative on its own, but also informative as to all its logical consequences. And then we have the following:


*

*An event horizon has further signification as being the place beyond which you, as an outside observer, cannot receive any information about events therein.

*From the above, this should mean that you also cannot receive any information that can be inferred as well as that is entirely direct.

*With the setup of a person holding a light as they fall into the black hole, if the light were to suddenly cut off as they crossed, you could infer from this the fact of their crossing.

*That means you could infer that an event within the horizon - "they have arrived on the inside" - has occurred.


Hence, the forces of the Universe must per logical necessity somehow conspire to cause that to fail to come to pass. And the way they do that is by infinitely prolonging the arrival of photons to any distant observer as seen on their proper time wristwatch as the photon source is closer and closer to the horizon, even though it gets through with no difficulties in its own proper time.

ADD (UE+1591.2427 Ms): Someone in the coments (Edouard) below took especial issue with my assertion that there's "no gravitational time dilation"; I should perhaps try to explain a bit more what I mean there. There clearly is, of course, an effect you can call as such, but not in the way that - and that's what I was getting at - some, especially perhaps less savant in this stuff, people might imagine it, at least in a "popular" sense. 
The "real" gravitational time dilation - which can be and is measured by sending a clock to a different part of the gravitating region and then bringing it back to yourself - is very much a real effect and, moreover it is not only analogous but mathematically exactly the same thing as - the "time dilation" taking place in the famous "twin paradox" of special relativity, where you send a craft out at high speed and then have it return, just happening in a curved space time manifold: both are a direct consequence of the anti-triangular inequality that must be satisfied within the time-like domain of any pseudo-Riemannian metric so that the number of hopping points between any two events is longest along an inertial path between them, i.e. one along which an object reproduces itself via its four-momentum alone with influence of no forces, and less so along other paths. Manoeuvering to and back from the horizon vicinity is a non-inertial manoeuvre just as the round trip of the twin, so it is shorter, so less action will be taken and a clock riding on it will record less time.
This time dilation is "real" in the sense I've been using because it is phrased entirely in local language, which means it depends on no arbitrary choices.
But the sense I am referring when I say that "gravitational time dilation is not real", what I am meaning is a somewhat different idea of imagining that the distant clock is, "while it is there", ticking away, "right now" at a slower rate, that you can give with some formula, for that requires you to have a notion of "now" extending out there by which you can measure it and note the fact of its "ticking", and depending on how you define that, you can enjoy great freedom (namely anything within the thickness of the "elsewhere" of where you are where it is will work) to assign it an arbitrary interval, because there is no natural choice of distant time coordinate, i.e. no natural simultaneity. That is, it makes no sense to say that "right now, that clock that we sent out to near the black hole is only ticking 1/100th as fast as were it here", because there is no such thing as "right now" outside your own self. It's just as valid to say it's ticking normally, or 1/10th the "normal" speed, or maybe (at least for a short time on your own watch) even faster than normal, because you can just make up coordinates that will do this. But even here, that said, you can still say that the absolute sense mentioned before does kind of intrude in that, however you make those coordinates, so long as it's a valid system, the rate at which it ticks in your coordinate system's coordinate time will of necessity have to be variable in a just-so manner so that once it returns to you, the number of ticks that the arbitrary-nows coordinates measured and that the clock actually reads, will ultimately coincide, for that is taking place in your local region and thus must be "objectively real".
A: It is not because of the doppler shift, but because of the gravitational time dilation. In the frame of an external stationary observer the infalling observer indeed never reaches the horizon. 
If both observers carry entangled quantum bits, and the external observer makes a measurement on his first, he also determines what the infalling observer will measure on his; should the infalling observer make his measurements at or after he fell through the horizon, the external observer has an infinite amount of time to make his measurement and thereby define what the infalling observer will measure on his.
A: My answer refers to an Einstein-Rosen black hole, which differs from the Schwarzschild black hole described in other answers, inasmuch as it is not a "vacuum solution":  It can contain matter.  (I believe the Schwarzschild variety can as well, but only through use of the Vaidya coordinate system, whereas the Einstein-Rosen variety has mainly been analyzed through Kruskal-Szekeres coordinates.)
In Nikodem J. Poplawski's inflationary "cosmology with torsion" (described in his many papers posted on Arxiv between 2010 and 2020), the idea of an "infinite time" required for passage over an event horizon relates to sequential decreases in the spatio-temporal scale of each of a possibly infinite no. of local universes, each materialized after the exhaustion of its nuclear fuel has left any unusually large rotating star susceptible to gravitational collapse, thru the locally intensifying effect of the gravitational field: All fermions spin, and this effect's transmitted by contact between the vastly larger-scaled stellar fermions and the newly-materialized ones, reversing and greatly accelerating the passage of many of the latter over their trajectories. (For the occurrence of these effects of torsion, all fermions must have spatial extent, which is required by the mathematically-complicated Einstein-Cartan Theory used by Poplawski: Although ECT meets all observational and experimental proofs of General Relativity, GR implies that all fermions may be "point-like".)
In Poplawski's cosmos, new and smaller fermions form a new, causally-separated "local universe" initially just inboard of the surface of the vol. occupied by the star before its collapse, which might as well be described as a temporal iteration extant within the duration of the much larger "parenting" iteration.  Because the same process could be occurring within it, and within each of the causally-separated regions which it might itself contain (without including any of them), the durations of time involved could be replicated infinitely on endlessly-decreasing scales, so that the reference to an infinity of time was physically appropriate in the hypothetical context of ECT.  
Even though ECT was developed through conversations between Einstein and Cartan in the late 1920's, and Einstein's overall reputation suffered some decline after he had lost the informal debate with Bohr over the EPR Paradox in the mid-1930's, ECT's assumption that fermions have spatial extent remained common in "pop-sci." texts well into the 1960's, when the spectacular technological achievements of quantum physics pushed it--perhaps only temporarily--onto the back burner.
Although Poplawski draws on ECT to provide an unusually complete mechanism for its realization, the possibility of cosmological down-scaling (possibly balancing expansion, and consequently allowing a past- and future-eternal process even under the strictures of 2003's Borde-Guth-Vilenkin Theorem) may also present itself in Penrose's "conformal cyclic cosmology" (which includes conformal rescalings, that preserve angles but not lengths), and, much less formally, in Wheeler's notion (repeated by Feynman) of quantum foam as possibly consisting of bubbles that might conceivably comprise causally-separated regions, at least until whatever collisions of them might occur.
