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Fusible numbers have been discussed here before. Other links: fusible numbers. OEIS A188545. You have an unlimited number of irregularly burning fuses that will nevertheless burn for exactly 1 minute. What is the smallest amount of time you can measure just over $0, 1, 2, 3, 4$ minutes?

The smallest fusible number above 0 is $0 + 2^{-1}$.
The smallest fusible number above 1 is $1 + 2^{-3}$. The smallest fusible number above 2 is $2 + 2^{-10}$.
The smallest fusible number above 3 is $3 + 2^{-1541023937}$.

Let fuse(0) through fuse(3) be $1, 3, 10, 1541023937$. Actually, that last number is under some debate, there are some papers arguing that it is much larger.

After wrangling with this, I'm wondering if there is a clever proof that fuse(4) and beyond are finite numbers.

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Ed Pegg
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    You just need to show that the fusible numbers are well-ordered. This is demonstrated here: http://arxiv.org/pdf/1202.5614v1.pdf – mjqxxxx Jun 11 '14 at 17:13

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