Yes, typically there will be a plateau. There's usually no way to guess exactly where the plateau will be, a priori; the only way to find out is to build larger and larger data sets and see what happens.
The size of the data set needed to reach a "plateau" is dependent on many factors, including the specific classification task you're trying to solve, the machine learning algorithm you're using, the set of features you've chosen, and maybe other considerations. For some tasks, a fairly small training set might be sufficient to reach the "plateau". For others, you might need an incredibly enormous training set -- so large that you'll never reach it in practice, so effectively you'll never hit a plateau in practice.
There's absolutely no way to answer your specific questions about how many images are needed for particular domains. I know you're hoping there will be some rule of thumb that'll help you predict how many images you need for some new classification task, but I've got bad news for you; there's not really any useful rules of thumb that I'm aware of. The only way to find out is to try it, or to read research papers or experience reports from others who have tried to solve the same task or a similar task.
It's also worth keeping in mind that the notion of a plateau should not be taken too seriously. Rather, you should think of it as diminishing returns. After some point, increasing the size of the training set will give diminishing returns in accuracy. It's not to say that there's necessarily a point where you get literally zero improvement; it's just that the improvement available from a larger training set might be extremely small, i.e., where the accuracy curve starts to become nearly horizontal, when graphed as a function of the size of the training set.
In the comments you separately talk about "the point when a classifier reaches 95% of human accuracy". That's not the same thing as a plateau; that's a totally different measure. For some classification tasks, humans are way better than any classifier we know of; we don't know of any machine learning approach that will get anywhere near human accuracy, let alone 95% of human accuracy, no matter how large a training set you have. For other classification tasks, machine learning can do better than any human -- so the classifier gets not just 95% of human accuracy, but maybe 200% of human accuracy. Comparing a classifier's accuracy to human accuracy is unrelated to the existence and location of a plateau (i.e., diminishing returns as you increase the size of the training set).