Sorry, can you clarify? Adding the template argument causes a crash on idone, but I have no idea why that happens. Moreover, the output on my computer is always b\n.
EDIT: Not sure if this is it, but after some digging I've found [0] on lifetimes of temporaries. As for the lacking template argument, presumably const char* is inferred and a temporary string is constructed in main?
So first of all, max("a", "b") expands to max<const char*>("a", "b"). It computes the maximum of two pointers, without looking at the characters at all. Comparing two pointers to unrelated objects is unspecified behavior, but in this case it usually returns the second pointer.
As for max<string>("a", "b"), it implicitly converts both arguments into temporary strings, then returns a const reference to one of them. Since it's a reference to a temporary, it becomes dangling when max returns, but you can assign it to a const string& anyway. When you try to access it, you'll get garbage or crash.
1) Replace max("a", "b") with max("b", "a"). Run it, look closely at the result, and be enlightened.
2) Replace max("a", "b") with max<string>("a", "b"). Run it, look at the result, and be enlightened in a different way.
As a small hint, your code invokes unspecified behavior, while my code invokes undefined behavior.
If this exercise doesn't put the fear of C++ into you, then I don't know what will :-)