The issue of infinite regress arguments came up in the comments to this post at "The Garden of Forking Paths"; and since I was intending to do a post on infinite regress arguments in the next several days anyway, given that they'll play a role in some of my Points in the Why I Believe in Free Will series, it has stimulated me to post it now.
What's wrong with infinite regress as such? Nothing at all. Infinite regress becomes a problem in only two types of circumstances:
1) The infinite regress arises in an attempt at explanation that involves the endless deferral of what actually explains the explanandum.
2) The infinite regress implies a contradiction given the nature of the regress.
An example of (2) occurs in Aquinas's ways to God, some of which use the impossibility of certain kinds of infinite regresses to prove the existence of something that's the sort of thing human beings call 'God'. Aquinas does not appeal to the impossibility of infinite regress as such; indeed, he is quite insistent that some infinite regresses are possible, which is a reason why he believes that we cannot demonstrate that the world had a finite past. What Aquinas does is argue that in certain cases, e.g., a regress of movers, or a regress of efficient causes, positing an infinite such regress implies the existence of something both unmoved and moved, or both caused and uncaused.
Interestingly, this is not how Aquinas's arguments are usually explained; they're usually explained as if they were (1)-type arguments. They are not. It seems likely that, given certain notions of explanation you can construct a sound (1)-type argument to go with any sound (2)-type argument involving any sort of dependence (which is what generates the contradiction in all the cases I have seen); but these are not the arguments Aquinas actually makes. And, while there is in certain cases a link between the two types, they are different types of argument, since (1)-type arguments yield practical contradictions, i.e., they set up a contradiction between what you are trying to do (explain) and what you are putting forward in trying to do it (complete deferral of explanation). But (2)-type arguments yield logical contradictions. Aquinas occasionally does run refutations based on practical contradictions (most notably against the Averroist doctrine of the intellect), but they are very rare; like all scholastics, Aquinas tends to look for logical reductio, not pragmatic retorsion.
So when an infinite regress is the sort found in (1) or (2), we are forced to achieve state, i.e., we have to terminate the regress in some way, or we are caught in a contradiction. It is the contradictoriness of certain regresses that causes the problem, not their infinity as such.