I went to Rebecca's Cave, an easy bike ride from our hotel, the Cayman Brac Beach Resort. Named after an 18 month old girl who died during a terrible hurricane, it has served as a hurricaine shelter for residents. It made me think about what it must be like to be on this tiny, isolated, wonderful island in horrible weather. Totally non-touristy, and worth a quick visit.
I also visited the Bat Cave, not far away. Unfortunately, no bats, and no Bruce Wayne. But it is a nice cave complex. Bats definitely live there, as evidence by piles of guano in little recesses along the cave roof. Very nice limestone formations. Bring a flashlight if you want to explore deep into the caves.
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