Will this program terminate for every Integer?
The correct answer is that this function does not terminate for all integers (specifically, it does not terminate on -1). Your friend is correct in stating that this is pseudocode and pseudocode does not terminate on a stack overflow. Pseudocode is not formally defined, but the idea is that it does what is says on the tin. If the code doesn't say "terminate with a stack overflow error" then there is no stack overflow error.

Even if this was a real programming language, the correct answer would still be "does not terminate", unless the use of a stack is part of the definition of the language. Most languages do not specify the behavior of programs that might overflow the stack, because it's difficult to know precisely how much stack a program will use.

If running the code on an actual interpreter or compiler causes a stack overflow, in many languages, that's a discrepancy between the formal semantics of the language and the implementation. It is generally understood that implementations of a language will only do what can be done on a concrete computer with finite memory. If the program dies with a stack overflow, you're supposed to buy a bigger computer, recompile the system if necessary to support all that memory, and try again. If the program is non-terminating then you may have to keep doing this forever.

Even the fact that a program will or will not overflow the stack is not well-defined, since some optimizations such as tail call optimization and memoization can allow an infinite chain of function calls in constant-bound stack space. Some language specifications even mandate that implementations perform tail call optimization when possible (this is common in functional programming languages). For this function, f(-1) expands to f(f(-2)); the outer call to f is a tail call so it doesn't push anything on the stack, thus only f(-2) goes onto the stack, and that returns -1, so the stack is back to the same state it was in at the beginning. Thus with tail call optimization f(-1) loops forever in constant memory.