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I am not into number theory at all -- is there a specific reason why some researchers spend enormous effort on calculating millions of digits in the decimal (or any other base) expansions of irrationals like pi? Are there known connections to applied fields such as cryptography, e.g.?

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Mostly, these calculations are taken as benchmarks for new supercomputers. I know more than enough digits by heart to compute the earth equator up to subatomics. Then again, the algorithms used are interesting in themselves (probably more than their output): Don't you find it intriguing that it is possible to compute the millionth digit o f$\pi$ in reasonable time without needing to calculate and store all previous digits? On the other hand, while it is widely believed that the digits of $\pi$ "look like random" no stats derived from billions of digits can actually prove such a conjecture. – Hagen von Eitzen Sep 27 '12 at 22:01
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@HagenvonEitzen, isn't that statement about computing digits of $\pi$ only about hexadecimal digits, not decimal digits? – lhf Sep 28 '12 at 0:01
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Does climbing Mt. Everest have an application? Are there known connections to applied fields such as geology? – Qiaochu Yuan Sep 28 '12 at 0:31

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