Posted links to all here:
Wikipedia has links to old ones below. The mechanical ones trip me out.
Another little-known use of analog is obfuscating trade secrets in ASIC's. One hardware engineer that used to post on Schneier's blog told us how most engineers and tools did digital with little understanding of analog or especially combinations of the two ("mixed signal"). So, he would put parts of the critical algorithms in analog with odd tie-ins to the digital side. Integrating their I.P. with digital tools was still easy but ripping it off required analog or mixed-signal experts spending a lot of time. A trick worth remembering.
Here are 2 photo I took of Eastern Block analog computers: https://twitter.com/flohofwoe/status/650324920862441472 (the grey one was from 1975, which was really late in the analog era, it was only used for educating students who had to work with older analog computers). And more impressions from someone's blog with much better image quality: http://blog.awsm.de/post/130463580578/vintage-computer-festi...
Also interesting: the Soviets built analog water computers starting in the late 1920s which could solve specific problems so fast that digital computers needed 50 years to do the same thing in comparable time: http://www.digitaljournal.com/article/338106
I once had two 555 timers, set up as oscillators, on a solderless breadboard, and they'd phase-lock just from inductive coupling if they were near the same frequency. Without bypass caps at each IC, they'd phase-lock even when the frequencies weren't close, at some ratio like 7:6. When the big power transistor in a 555 switches, it introduces a transient, which gets into the input signals. Fun to watch on a scope.
While you can build analog differentiators, they're so noise-sensitive that they're almost useless. Combined differentiators and low-pass filters can work, but a pure differentiator tries to track the slope of every tiny noise event. Analog computing is usually all integrators, summers, and multipliers, as they did here.