Yahoo! Mail and Flickr have suprisingly big user bases, and I don't think it's a stretch to fortify those into strong contenders.

I view Mail as 2nd only to Gmail, especially among the non-techies. The new interface is pretty decent (my ex-coworker loved it, and he's an interaction designer). I can see it having a big presence if they improve it.

They just released a new Flickr app, which is actually pretty nice. Unfortunately the target user that they've focused on has been the hardcore high quality photographer, not the everyday Instagram types, so it feels weird to upload Instagrammy photos onto there.

But even if Yahoo Mail is a great product, I use Gmail for its integration and proximity to other Google products which Yahoo can't compete with: Docs, Maps, Chat, etc.

I think you're ignoring a huge population that doesn't think about things like "integration" and "proximity" and "google" vs "yahoo". Yahoo! has a large user base already, even thought Google is already tightly.

tightly integrated*

Also another big thing for Yahoo!: Fantasy Sports.

YES! Yahoo Fantasy defined my high school experience.