I don't think it's easy to put a limit on the maximum achievable speed (other than $c$), without constraining the problem. There is no obvious speed, above which gravity assists can't help, if you can plan the interactions appropriately.
There are things that will make this tricky though. For example the large relative speed that bodies will likely be encountered when moving fast, and the necessity to align exit velocity with the velocity of the assisting body, you are quickly going to run into the problem of needing near infinite accelerations.
I.e. skimming event horizons of progressively smaller black-holes.
This may bump into some other hard physics limit (like a black hole that small existing for long enough to complete the interaction). However I don't think there is a nice/simple statement that governs this sort of thing. Further You're starting to blur the lines of what constitutes a gravity assist when you start considering these sorts of things.