Recursion depth limit is really a memory limit, so you just need someone to come and install more RAM when you run out in order to obtain de facto infinite memory, hence infinite recursion depth.
Really, the crux of the matter is that you can have a function call itself an unbounded number of times in C (i.e. unbounded recursion). That you eventually run into a stack overflow is a limitation of the machine, not of the language; the definition of C does not prescribe a maximum number of recursive calls. This, together with conditionals, makes the C language Turing-complete.