How can classical black holes even exist? I am not asking about the event horizon, but the actual black hole itself and I am asking this question based on the following thought experiment:


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*Imagine the hypothetical process where you have a black hole being formed out of three particles by applying a form of external pressure.

*At some point in time, one of these three particles (particle A)  is supposed to be so compressed that it fits inside its own event horizon. Lets call this event to "transition".

*Immediately before particle A transitions, time at its surface is almost infinitely dilated. It would, as seen from our distant frame of reference, seem to take an infinite time for the surface of that particle to actually cross its own event horizon - as observed externally.

*The two other particles, also being external to particle A (albeit much closer)  would also see this as something that takes almost infinite time. Due to their own time dilation, they will see it happening faster than some remote observer - but still the time dilation should approach infinity. Particle C will never see particle B merge with particle A, and particle B will never see particle C merge with particle A.
Now, I do understand that multiple particles together can create a gravitational well, from which neither particle can easily escape and that this will look like a black hole for a distant external observer.
But if you look closer, each of those particles should never be able to observe parts of themselves actually crossing the event horizon of another particle? 
 A: 
At some point in time, one of these three particles (particle A) is supposed to be so compressed that it fits inside its own event horizon. Lets call this event to "transition".

I believe that this is the source of your confusion. This is not a good description of the formation of a black hole. 
The classical solution describing the formation of a black hole is called the Oppenheimer-Snyder spacetime. It models the collapse of a spherical pressureless cloud of dust, so there is no pressure at any time in the solution. Although that is obviously an egregious simplification, it turns out that the basic features of this simple analytical model of collapse are shared by more accurate numerical models. 
The key for initiating collapse and formation of an event horizon turns out not to be pressure at the center, but simply having enough mass inside the Schwarzschild radius. Even very low densities will be sufficient for collapse if the total mass is large enough. 
When sufficient mass is accumulated, the event horizon forms at the center of the cloud and rapidly expands to cover the entire cloud. The density may still be low throughout the cloud, although it will rapidly increase at the center where the singularity gains mass. But the singularity is always inside the horizon. 
A: Many "bouncing cosmologies" (by Lee Smolin, Nikodem Poplawski, and others) substitute the formation of a new "Local Universe" for the singularity mentioned in an earlier answer: Poplawski goes into the most detail, in several papers on the Arxiv website that were written as recently as 2018. These cosmologies generally have the advantage of balancing a contraction (which begins when the star runs out of pressure-producing fuel) against expansion, thereby allowing a universe eternal to the past, which is otherwise impossible (for geometrical reasons described in the Borde-Guth-Vilenkin Theorem).
I have to clarify that the universe eternal to the past would be an ensemble, or set of iterations, of local universes. There is ample astronomical evidence for black holes, including the circular orbits of many lone stars whose binary partner has become one. Most stars are in binary pairs. 
I think the OP may be thinking of microscopic black holes, once an idea of Hawking's, for which there is little evidence in our currently-observable region. (The spatial scale of objects outside it is, of course, entirely hypothetical.) 
I believe he may have been misinformed on two points: First, that BHs result from external pressure rather than a lack (as I pointed out earlier) of internal pressure, and, second, that the event horizon appears somewhere inside the collapsing object and stays there--it doesn't; it propagates outward from the center of the collapsing star. The ideas of the collapse taking forever are correct only in the inclusion of further BHs within the local universe in formation, more BHs within them, etc., and derive from the "past eternality" permitted by the balancing of such contraction against expansion.  
That last statement sounds so outrageous that I feel obligated to back it up with a little math, visibly provided by Vilenkin in his 2013 paper "Arrows of time and the beginning of the universe", a critique of the bouncing cosmology that Carroll and Chen had devised in their own paper "Spontaneous inflation and the origin of the arrow of time":  Vilenkin concludes that "an infinite Cauchy surface with random initial conditions will generally produce inflating regions in both time directions", which would, however, "be surrounded by singularities, and...have singularities in their past and future", while the current Wikipedia article "Gravitational singularity" describes singularities as having the "infinite density" and "infinite temperature" I'm eliciting in that statement.  (Carroll and Chen don't particularly set their cosmology within any black hole, but Poplawski provides a mechanism which would get a local effect going that could easily--even to inhabitants only average-sized on an infinite range of scales--resemble "our" Big Bang sufficiently to be, for them, as observationally indistinguishable from it as our own appears to be for us.  All of this could happen within space appearing to be black, subjectively deep, and laden [by whatever Hawking radiation would've evaporated previous BHs] with enough potential for point-like bosons to get just such a shebang rolling again, in any region that had been rather quiescent for a while.)
The "event horizon" is, since remarks by Hawking in 2014, sometimes referred to as an "apparent horizon", which leaves the door open to whatever evidence might eventually substantiate a beginning for a multiverse.  Such evidence might include changes in the CMB data, although our current CMB data, described in the Arxiv paper "Non-parametric reconstruction of an inflaton potential", is consistent with Poplawski's theory.
In case anyone might conclude that I'm implying that a multiverse might consist of a quasi-infinite collection of sub-microscopic, toy-like versions of our observable region, I have to say that that's not necessarily my aim. Although such a representation of it might be consistent with such a scale-invariant theory as GR (especially given Einstein's comments about the speed of light, that can be seen in the 1st paragraph of Section 27 of Wikisource's online version of his 1916 popularization "Relativity: The Special and General Theory"), the effect of time dilation, in the intense gravitational field surrounding the center of any large star undergoing gravitational collapse, might balance any reduction in the spatial scale of material and energetic objects (including both subatomic particles of the star itself and new particles, made real by their separation from partners in virtual pairs, drawn into the collapse by tidal effects) arriving there, while the reversibility of the length contraction in the material ones, during their subsequent outward acceleration from contact with the stellar fermions, might combine with it to leave (or, at least, enhance) that impression of spatial expansion which characterizes our Big Bang.    
A: There's a reason they're called black holes - they're not observable. 
In physics, if it's not observable it doesn't exist. 
Further, GR has only been tested in weak field approximation - it's behavior in presence of strong fields is pure speculation. 
And for the General Theory of Relativity (GR) to have a physical singularity would be devastating.
