Globular clusters are apparently very very old, and the density of these clusters appears to increase as one approaches the center of a cluster. Orbits are bound to be chaotic, since there is no particular orbital plane, unlike a spiral galaxy. From tidal effects alone it seems that over time many of the stars in the middle ought to have merged, forming new stars of greater and greater size. Eventually one should have seen supernovae occurring inside these clusters, or at least so it would seem, and there ought to be black holes in some of them. However, it appears that this does not happen, and the stars in these clusters do not merge. What is keeping this from happening?
There are three things I'll try to explain in my answer. First, globular clusters actually take a long time to evolve, on the order of $10^8$ to $10^9$ years. Second, many encounters between a binary and a wandering star can eject the wanderer with only a small tightening of the binary, thereby depleting the core. Third, it's currently contested that some globular clusters actually do have black holes at their centres.
The rough timescale for the evolution of a globular cluster is the crossing time. That is, the time an average star takes to traverse the whole cluster. The Wikipedia entry for globular clusters says "the mean value is on the order of $10^9$ years". So it would take some time to get a really big black hole to build up.
It's true that there are many interactions between stars in the dense core. However, a lot of these lead to the formation of binary systems rather than collisions or mergers. When a third star interacts with a binary, a common outcome is that the stars in the binary move closer together and energy is conserved by ejecting the third star at high speed. I recently saw a seminar where the speaker did a detailed calculation of how much "radiation" this produces and it turns out quite a lot. Something like, if a binary forms, it can eject 20 or so stars before the binary merges into a single star.
Finally, don't forget that people have already claimed to have evidence for black holes of a few thousand solar masses at the centres of M15 and $\omega$ Cen. They've also suggested such black holes at the centres of globular clusters around other galaxies. Mayall II is one candidate, orbiting Andromeda. The recent intermediate-mass black hole candidate HLX-1 seems to have a population of stars around it, so it might be a globular cluster (but it might also be the stripped core of a galaxy that recently interacted with ESO 243-49).