[updated after a new suggestion]
Here are some of the Everests for Computational Cognitive Modeling. Some people call them AI-complete. That might not be the best term, as it extends the notion of NP-Completeness, which is a precise, formal, mathematical notion, into a very blurry territory.
Anyway, I've put them from easier to harder...
- Cracking Captchas: not to spam people, but, as Everest climbers say, "Because it`s there"
- Computer Go
- Speech recognition: for example, being able to listen to a podcast and transcribing it)
- Bongard Problems: solve them and you've solved the general pattern-recognition problem, which means, at least to your blogger...
- face/picture recognition-->suggested by pfctdayelise in the comments (see also my response)
- Machine translation
- Passing the Turing Test, or the Total Turing Test.
Do you have another problem that's missing from this list? I would appreciate additions and suggestions in the comments.