I am describing an event that lasts for more than 68 continuous hours, and I want to describe it as a "multi-day long" event, but I don't know where to put the hypen(s) in that phrase.
If you're creating a noun phrase, drop the "long" and simply call it a "multi-day event".
Alternately, consider "the event spans multiple days" or (less elegantly) "the event is multiple days long".
Technically speaking, it's not wrong to hyphenate it as a multi-day-long event.
You can have a day-long event, so multi is simply added to the start of the compound adjective. However, that does look a little strange.
In addition to the other solutions, you can also call it a days-long event. This, too, is a bit strange, but still correct.