It has been widely stated that Arnold Schwarzeneggar - and indeed Michigan governor Jennifer Granholm - would be barred from a run for the US Presidency by virtue of not being a natural born citizen of the US. It is assumed it would require a constitutional amendment to facilitate this.
There is an oft-overlooked exception to this, as none of the early presidents were natural born citizens. Someone who is a citizen at the time the constitution was adopted is also eligible to run. I am not aware of any Supreme Court rulings on this, but it is reasonable to suppose that this does not only refer to the original adoption, but to its adoption by any particular state. For example, Senator Daniel Inouye was a citizen of Hawaii at the time it first became a US state, and was elected as that state's first congressman. Is he eligible to run for president? Schwarzeneggar's native Austria seems unlikely to adopt the US constitution, but if he were to become a citizen of Puerto Rico, and that territory were to subsequently become a state . . .