Several years ago I was one long-haul round-trip away from AA EXP based on EQP. I was already planning to take an international trip but the deep-discount economy fare that I was looking at only earned 0.5 EQP per mile, which would not have earned enough to get me to EXP. I booked in a higher fare class that would earn 1 EQP / mile for ~$500 more and it worked out great. As I recall, AA.com only showed the lowest-cost fare class, so I had to call in, explain what I wanted, and book by phone. The agent was confused why I wanted to pay more than the lowest available economy fare, but once I pointed out the different EQP earning rate, she immediately understood.

I don't think my situation will recur, now that (I believe) AA has gotten rid of qualification through EQP.