Does a trigger that says 'at the beginning of your next upkeep' wait until your very next upkeep even if you have multiple turns?

The phrase 'at the beginning of your next upkeep' creates a delayed triggered ability that is set up to trigger exactly once — the next time your upkeep step begins after the effect was created. This is governed by CR 603.7, which covers delayed triggered abilities, and CR 603.7b, which specifies that a delayed trigger waits until the next occurrence of the specified event.

Because the trigger is looking for your next upkeep, it counts only upkeep steps belonging to you, the player who owns the trigger. If an opponent takes an extra turn in between, their upkeep steps are irrelevant — the trigger patiently waits and will fire at the start of your very next upkeep, regardless of how many total turns have elapsed.

Similarly, if you somehow take extra turns yourself, the trigger fires at the start of the very first upkeep you reach after the delayed trigger was created, then it expires. It does not linger to fire again on a second upkeep of yours (CR 603.7b: unless the ability says 'each' upkeep, it triggers only once).

Example: During your main phase, you cast a spell that says 'At the beginning of your next upkeep, draw a card.' An opponent then takes an extra turn with Time Warp. During that extra turn's upkeep, nothing happens. When your next turn finally starts and your upkeep begins, the delayed trigger fires and you draw a card.