Usually programs hit the limits of float32 or float64. A float64 has a around a 10 to the minus 16 precision to play with (53 bit significand) so you can hit the zoom limit with little effort.

There are various techniques to go further (obviously one is just to use arbitrary precision arithmetic, another is for example perturbation theory[1])

And there are plenty of "infinite" (or at least some arbitrarily huge level) zoom viewers. You can find videos of zooms going into 10 to the power of thousands. The deeper you go the longer the calculations take.

[1] https://en.m.wikipedia.org/wiki/Plotting_algorithms_for_the_...