How many critical points can a function have?
It depends on the type of function.
A polynomial can have zero critical points (if it is of degree 1) but as the degree rises, so do the amount of stationary points. Generally, a polynomial of degree n has at most n-1 stationary points, and at least 1 stationary point (except that linear functions can't have any stationary points).
Trigonometric functions have infinitely many such points, unless the domain of the function is restricted. These points lie at x=0+n(pi) for f(x)=cos(x) and x=(pi/2)+n(pi) for f(x)=sin(x). If the period changes, so do the stationary points, but there still exists an infinity of such points.
A lot of other basic functions can't have any stationary points at all. The hyperbola of f(x)=1/x has none. The exponential function has none. The logarithmic function has none. The square root function has none.
Many other functions aren't "pretty" and so further analysis is usually required to find out how many points these functions have.