Only one type, but there's not limit on subtypes.
Frankenstein's Monster (Adam) would be a Construct. A proposed Undead Construct would be a Construct that was killed, and then later brought back to a sembalance of life. Doesn't work because a construct isn't alive in the first place; no Con score. So Adam, along with all his lesser-than-he, unintelligent flesh golem ilk, are Constructs even though they're sewn together from dead person bits.
An Outsider Dragon, like say a green dragon with the fiendish template, would be an Outsider. The reason why being covered in the fiendish template section where it lists the changes to type. Overall not that great a deal for the dragon - hit dice drops to d8, but that weak ref save does get shored up a bit.
New type supercedes old type. The old type becoming a subtype ("Augmented X") in the case of applied templates and changes, such as the kind you get from advancing 10 levels in certain prestigue classes. In the case of inherrited templates - a template you're born with - the base creature's original type doesn't become a subtype. At no point did you suddenly become something more than what you were - you've always been that way.
As for huge problems in house ruling critters to have multiple types... I could see it being problematic. Your type determines what benefits you gain from any racial hit dice: your ba, saves, skills, etc. If you're tossing multiple base types on a critter, which type's criteria do you use? Do you average both out? Do you just get the best of everything (which leads to the question of the heck is the point in mucking about with types if all you're going to do is just take the monster and 'turn it up to 11')? So on and so forth.