$1 invested for 1 year at 100% annual interest. Compounded once: $2. Compounded twice: $2.25. Compounded a hundred times: $2.7048. As the compounding frequency goes to infinity, the answer is exactly e.
100% annual rate compounded yearly, $1 becomes $2. Compounded twice yearly (50% twice), it becomes (1.5)² = $2.25. Compounded n times, it becomes (1 + 1/n)ⁿ. Bernoulli noticed this number had a limit as n grew — but he didn't compute it. Forty years later, Euler computed it to 23 digits and named it e.
e^r in one year, and to e^(rt) in t years. Sliding the n-slider from 1 to 365 watches the discrete approach the continuous; the two converge faster for small r and slower for large r.