A very well-known irrational (and transcendental) constant. It's also called Euler's number after Leonhard Euler, who made the constant famous.
It can be defined as the limit of (1+1/x)^x as x tends to infinity. e can also be calculated as the sum 1 + 1/1! + 1/2! + 1/3! + ... or the sum of the reciprocals of factorials. In fact ex can be calculated as 1 + x/1! + x^2/2! + x^3/3! + ...
The derivative of the function ex is ex (the same function). This means that the slope at any point along the y=excurve is equal to the y-coordinate of that point. In particular, the slope at x=0 is 1. In fact e can be defined as the number a such that the slope of y=ax at x=0 is 1.
e appears many, many times in calculus, and appears a few times in googology, such as when studying infinite power towers, Stirling's approximation on large factorials, and the definition of Skewes' number.