It triggers for each permanent that untaps, so if ten permanents you control untap, Mesmeric Orb’s effect puts the top ten cards of your library into your graveyard.

Mesmeric Orb triggers once for each permanent that untaps, so the cards are put into your graveyard one at a time.

If permanents become untapped during the untap step, Mesmeric Orb’s ability will trigger that many times. However, since no player gets priority during the untap step, those abilities wait to be put on the stack until the upkeep starts. At that time, all the “beginning of upkeep” triggers will also trigger. Those abililties and Mesmeric Orb’s triggers can be put on the stack in any order.