I've heard mentions of people using tertiary, quaternary, quinary, etc fermentations. Does this really provide any significant benefit beyond secondary?
I think this same answer applies here:
The purpose of n-ary isn't more fermentation, but aging, clarifying, etc.
So for me, no it doesn't provide any additional benefit. The only time I use a secondary vessel is for fruit additions or extended aging. I've never found a need for a third vessel.
I've heard of having a tertiary phase if you add fruit (etc.) to the beer during the secondary, to help precipitate out the fruit particles before bottling.
I can't get a pumpkin ale clean with only a secondary for the life of me. Next batch I plan on using a tertiary. Just be careful with splashing, I've been careless and oxidized the hell out of an otherwise good beer before.