Can different optimization levels lead to functionally different code?

Is it possible to write code which behaves differently depending on which optimization level it was compiled with?

Only if you trigger a compiler's bug.

EDIT

This example behaves differently on gcc 4.5.2:

void foo(int i) {
  foo(i+1);
}

main() {
  foo(0);
}

Compiled with -O0 creates a program crashing with a segmentation fault.
Compiled with -O2 creates a program entering an endless loop.

The portion of the C++ standard that applies is §1.9 "Program execution". It reads, in part:

conforming implementations are required to emulate (only) the observable behavior of the abstract machine as explained below. ...

A conforming implementation executing a well-formed program shall produce the same observable behavior as one of the possible execution sequences of the corresponding instance of the abstract machine with the same program and the same input. ...

The observable behavior of the abstract machine is its sequence of reads and writes to volatile data and calls to library I/O functions. ...

So, yes, code may behave differently at different optimization levels, but (assuming that all levels produce a conforming compiler), but they cannot behave observably differently.

EDIT: Allow me to correct my conclusion: Yes, code may behave differently at different optimization levels as long as each behavior is observably identical to one of the behaviors of the standard's abstract machine.

OK, my flagrant play for the bounty, by providing a concrete example. I'll put together the bits from other people's answers and my comments.

For the purpose of different behaviour at different optimizations levels, "optimization level A" shall denote gcc -O0 (I'm using version 4.3.4, but it doesn't matter much, I think any even vaguely recent version will show the difference I'm after), and "optimization level B" shall denote gcc -O0 -fno-elide-constructors.

Code is simple:

#include <iostream>

struct Foo {
    ~Foo() { std::cout << "~Foo\n"; }
};

int main() {
    Foo f = Foo();
}

Output at optimization level A:

~Foo

Output at optimization level B:

~Foo
~Foo

The code is totally legal, but the output is implementation-dependent because of copy constructor elision, and in particular it's sensitive to gcc's optimization flag that disables copy ctor elision.

Note that generally speaking, "optimization" refers to compiler transformations that can alter behavior that is undefined, unspecified or implementation-defined, but not behavior that is defined by the standard. So any example that satisfies your criteria necessarily is a program whose output is either unspecified or implementation-defined. In this case it's unspecified by the standard whether copy ctors are elided, I just happen to be lucky that GCC reliably elides them pretty much whenever allowed, but has an option to disable that.

Floating point calculations are a ripe source for differences. Depending on how the individual operations are ordered, you can get more/less rounding errors.

Less than safe multi-threaded code can also have different results depending on how memory accesses are optimized, but that's essentially a bug in your code anyhow.

And as you mentioned, side effects in copy constructors can vanish when optimization levels change.