However, you could use phrases to create a compound-like effect, depending on the context:
* President-elect (for someone who has been elected but not yet sworn in)
* Vice-president
* Acting President (for someone temporarily holding the office)
* Former President
* Presidential candidate
* Presidential election
You could also use adjective-noun combinations:
* Presidential term
* Presidential campaign
It's important to remember that language evolves and new compound nouns can emerge over time.