100 % accuracy is what the algorithm wants to achieve (in the most basic case), so that's your asymptote. If your precision increases logarithmically e.g. as 1-1/log(x) you get arbitrarily close to 1, which is the definition of an asymptote I think.

Okay, yeah you make a good point. Strictly speaking a logarithmic function does not imply a convergent series. To be more precise, what I'm talking about are slowly varying functions. These are functions which don't converge but which have similar characteristics to convergent functions (namely, extremely slow rate of change).

Since you made the correction you're probably already familiar with these, but for the benefit of others: https://en.wikipedia.org/wiki/Slowly_varying_function

It really slows down though. Log(n) indeed goes to infinity but if you numbered all the atoms in the universe and pass them through the function, the last one would be ~80.