What real-world problems have you solved with Extended Events?
There are TONS of theoritical ones but I'll stick to 1 that I used by using extended events, then another that a tool I demo'd found with extended events:

Had to build a system that met the SISR and PCI tier 1 requirements for a tier 1 telco.  They required us to log every SA log in.  Using extended events was a lot more lightweight and easy to manage than having to worry about profiler always being on.

2
-TempDB Contention.  A query would run 1 second, half a second, 30 seconds, it was all over the place.  The tool used extended events to break down the wait types and I saw PAGELATCH_SH was really high.  Turns out tempdb did not have enough data files to optimize all the CPUS (32 cores) that the server had running and we had tempdb contention.  Using the wait stats per query really helped figure that issue out fast.

-Was migrating a back office SQL cluster to a different data center, trying to keep downtime to 1 minute or less.  While testing the new data center I would get major timeouts.  Timeouts were usually caused by sql server however looking at the wait type of NETWORK_IO we were able to trace it to a firewall between the DCs that was pegged at 100% CPU dropping packets faster than monica lewinsky.  Yea that's right, my last example had Monica Lewinski lol.