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It is interesting to look for things that turned out to be much more useful than initially thought, but I think you ought to look for reasons that you know for studying Lychrel numbers instead of hoping that more will come in the future. It seems to me like the primary motivation is that this is a simple question that seems like it should be easy to answer, but apparently isn't, so by searching for the answer, we may come to understand the integers better.
This actually seems to be a non-example. Conic sections were apparently first studied by Menaechmus in the 4th centure BCE. We're not sure what his motivation was, but he definitely used them in his method of doubling the cube. Some speculate that this problem led him to discover conics; others suggest that he was prompted by the fact that the tip of a sundial traces a hyperbola on any given day (outside the Arctic circles, anyway). In any case, it looks like conic sections had applications as soon as people knew about them.
