Euler's number (≈ 2.71828...) is related to compound interest.

Consider making a loan of 1 BTC that yields 100% in a given time period. At the end of the loan term, you would receive your principal (1 BTC) and interest (1 BTC) for a total of 2 BTC.

What if the debt compounded twice, rather than just once? The 1 BTC would yield 0.5 BTC for the first half of the loan term. This 1.5 BTC would then yield 0.75 BTC for the second half, for a total of 2.25 BTC at the end of the loan.

Thus, increasing the rate of compounding increases the final yield. If the compounding is done continuously, where interest is being accumulated at all times, you would receive 2.71828... BTC at the end of the loan term.

That being said, don't loan out your bitcoin, ya dingus!