As others have noted, every state in the state space will eventually be explored and so the appearance of convergence is a little misleading. Eventually the richest person will lose all the money and someone else will take that place.

An interesting question is why we happen to see the system in such an unequal state when we look at it after e.g. 10000 iterations. The reason for this, I believe, is entropy. There are just vastly more ways to distribute the money unequally than equally (thinking of these as macro states). For example, there is only one (macro) state where everyone has $100, but there are 100! states where the money is distributed in a perfectly unequal way (e.g. 0, 1, 2, ..., 99, 5050 or any other way where all the people can be distinguished). So, if you randomly peek at the system at any given time it'd be very unlikely that you'd see it in any way other than a highly unequal state. To be fair, this argument neglects the dynamics (perhaps the transition probabilities to those states are sufficiently small that the n! multiplicity is watered down, although I suspect this is not the case.)