Why doesn’t gravity break down in a large black hole? By popular theory gravity didn’t exist at the start of the Big Bang, but came into existence some moments later.  I think the other forces came into existence a little latter.
When a black hole crushes matter to a singularity (infinite density), at some point shouldn’t the forces cease to exist including gravity?
Kent
 A: You are right. Both the Big Bang and a black hole are what is called a space-time singularity and the physics of these object is, to put it simply, poorly understood. We know a lot about a black hole seen from outside, or about what happened some ridiculously small after the Big Bang, but it is unknown if the laws of physics as they are currently understood cease to be valid before the Planck epoch ($10^{-43}$ seconds after the Big Bang).
The same applies for a black hole: laws of physics describe extremely well what happens outside the black hole, and, to an extent, even in proximity of the center of a black hole but the closer you get to the singularity, the higher the curvature and we simply don't know if Einstein field equations (the cornerstone of General Relativity) are valid for higher curvature, or are just a low-curvature approximation or the exact laws.
A: A black hole is in many ways like a time reversed version of the Big Bang. If there were several stages of symmetry breaking as the Big Bang evolved, then it's certainly possible that an observer falling into a black hole would see the symmetries restored as they approached the singularity. This wouldn't really mean gravity "breaks down" to use your phrase. It would mean that around a Planck time before you hit the singularity gravity would unify with the other forces. What exactly that means no-one knows as there's no accepted theory to describe matter under those conditions.
There are some objections to this: for example a black hole (probably) isn't a time reversal of the big bang. The Physics FAQ has a good article on this at http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/universe.html. Also the Weyl curvature is (probably) high as you approach the singularity of a black hole and the Weyl curvature near the Big Bang was (probably) low.
The main objection is that this is a somewhat fanciful question and it's currently impossible to put it on any sort of quantitative basis. Fun to chat about over a drink but I suspect you'd struggle to get papers on it accepted anywhere reputable.
A: During the phases of dying stars, forces of physics are overpowered by the crushing implosion of matter, until basically only gravity remains in the densest bodies (black holes). Why couldn't there be an impact so great that for a "moment" gravity itself "ceases" to exist, thus creating a big bang, and unlike a dying star which leaves a core, leaving a void instead... 
