Is space stretched with no limits by a black hole? Some depictions of black holes show space being warped into a singularity, with no end, e.g. as pictured below. Moreover, in Cosmos, Neil Tyson speculates with the possibility that Black Holes contain other "universes". I am not sure about what to make out of this, but it made me wonder:


*

*Can space be stretched with no limits by a blackhole? 

*If there is a limit, is there a way to quantify how much space the black hole "stretches"?

*Is the amount of "space stretching" (e.g. "extra space") determined in any way by the mass or volume of the blackhole?



 A: It's a difficult question to answer because in relativity, distance and volume are coordinate-dependent quantities.  Even without a black hole, special relativity allows us to distort distances (Lorentz contraction) to an arbitrary degree by moving at speeds close to lightspeed.
That being said, if we limit ourselves to Schwartzschild coordinates (which basically represent how an observer hovering at a distance from the black hole sees things), there are some interesting facts to examine.
First, the coordinate distance to the event horizon is indeed greater than would be expected in flat space, meaning if you hovered over the black hole and let down a tape measure, you'd need a longer-than-expected tape measure, by a factor of over 2, to reach the event horizon.  This means the volume of the region around the hole (outside the event horizon) would be greater than that contained in a region of flat space with the same circumference.
Diagrams like the ones you posted are basically slices of the spacetime at a constant Schwartzschild time coordinate, so the relationship of distances and volumes they depict is qualitatively accurate.  For objects like neutron stars that are close to being a black hole but not quite, there is also significant additional distance and volume, but less than for black holes.
On the other hand, if you freely fall into a black hole then you hit the event horizon and then the singularity in a finite time from your point of view.  There are other coordinate systems, namely Lemaître and Gullstrand–Painlevé, that describe the spacetime "from the perspective of falling observers" in some sense.  If you measure the volume of a constant-coordinate-time slice in those coordinates, you'll get something time-dependent (for Lemaître) or equivalent to flat space (for G-P).  This just goes to show that the "volume of space" in a curved spacetime region really isn't a well-defined quantity.
A: The OP asks is space stretched with no limits by a black hole. The answer to that is no. I think the typical way one would think of space is from some co-moving frame or similar, instead of from some arbitrarily accelerated frame, like when hovering outside a black hole using lots of rocket engine power. In an accelerated or high velocity frame space appears to stretch whether you are actually in flat space or not, as Nathan points out. Accelerated or high speed frames are not good platforms for measurement.
So if one admits that flat space as viewed from a highly accelerated high velocity frame is still flat, then a black hole is nice and behaved near the event horizon too, and space is not at all stretched without limits. In fact supermassive black holes have very well behaved horizons. The event horizon is reached in finite time over a finite distance. 
When one probes deeper - past the point of no return, one finds a finite length path to the singularity in finite time. So the trip to the singularity is not infinitely long - space has not been infinitely stretched. At the singularity GR breaks down, so not much can be said about whether space is stretched with no limits at that point. 
A: Although there is apparently no bound to the calculated proportionate stretching as you approach "the singularity" when viewed in Schwartzhild coordinates (albeit the integral remains finite), there is a major problem with this approach.   This concerns the development of the singularity: the Schwartzchild picture assumes an equilibrium condition.  This equilibrium is calculated from the perspective of an external observer (who would of course not be able to observe).   According to my elementary theory, if we also consider the external observer's time-frame, the calculated equilibrium is in fact unstable; it is capable of existence, but it could not actually form. Imdeed, if I calculate what is happening to mass that is not already in the singularity, we find that it is moving towards the event horizon, regardless of whether it is inside the event horizon or not.  On this basis, black holes that are formed via mass collapse will not contain singularities (even in the Schwartzchild co-ordinates); Primordial black holes could in principle have singularities (in Schwartzchild coordinates), but external observers could never know this.  (Cf Shrodinger's feline?)
