It partly depends on your goal. Do you just want to best estimate the constant asymptote? Or, are you looking for some function that will approximate the entire curve? The latter is of course far more difficult, since there is no reasonable model that will fit any possible such relationship, especially when it appears that your curve increases smoothly, then suddenly flattens out, with a virtual break in the slope at one point in one of the examples you show. Worse, at the bottom end, it looks like there is some garbage, SOME of the time.
If your goal is to just find the best possible constant asymptote estimate, then it is simple enough. Pick the set of points at the upper end where it appears to be essentially constant, then take the mean of y over that interval. The mean would be a good estimate of a constant function over an interval, the best estimate, in fact. You could be more sophisticated of course, in lots of ways, but that is simplest, and should be entirely adequate.
If your goal is really to get the entire curve fit smoothly, then I might recommend a smoothing spline, as most general. But a smoothing spline will not understand that the upper end of the curve approaches an asymptote. You could build that information into the spline model, but again, that will take more effort. And of course, splines have real problems with transitions from a high slope region into a constant slope region. You will always find ringing there near the transition, unless you use a spline model that includes a monotonicity constraint. Again - that will be more difficult to deal with.
Anyway, your question seems to be how to find a best estimate of the asymptote, which as I said, is not that difficult.