What determines whether a constexpr function is a constant expression?

Since you're not calling f in a constant expression, your question is asking if the compiler is required to diagnose that f can't be called in a constant expression, based solely on its definition.

The requirements on the definition of a constexpr function are enumerated here:

The definition of a constexpr function shall satisfy the following requirements:

(3.1) its return type (if any) shall be a literal type;

(3.2) each of its parameter types shall be a literal type;

(3.3) it shall not be a coroutine;

(3.4) if the function is a constructor or destructor, its class shall not have any virtual base classes;

(3.5) its function-body shall not enclose

(3.5.1) a goto statement,

(3.5.2) an identifier label,

(3.5.3) a definition of a variable of non-literal type or of static or thread storage duration.

As can be seen, the definition of f does not violate any of the requirements in the list. So a compiler is conforming if it chooses not to diagnose this.

As pointed out in aschepler's answer, constexpr functions like f that can't be called in a constant expression, but are not diagnosable as such, are considered ill-formed-no-diagnostic-required.

Both programs are "ill-formed no diagnostic required", per [dcl.constexpr]/6:

For a constexpr function or constexpr constructor that is neither defaulted nor a template, if no argument values exist such that an invocation of the function or constructor could be an evaluated subexpression of a core constant expression, or, for a constructor, an evaluated subexpression of the initialization full-expression of some constant-initialized object ([basic.start.static]), the program is ill-formed, no diagnostic required.

It's a bit strange that gcc just fails to notice the issue with the second program, but it's still conforming.

Note a diagnostic would be required if f were used in a context that actually requires a constant expression, for example constexpr int n = f();.

Some things are never permitted in a constexpr function. These do require a diagnostic (typically an error message), even if the function is never used in a constant expression - see cigien's answer. But the programs in the question don't violate any of these stricter rules.