Pick a set of integers, a set of base integers, and a set of operations, and start your watch to see how many integers you can make from exactly one each of the base integers and the operations. Here are 0 through 50 using the base integers 1, 2, 3, 4 and +, -, *, exponentiation, parentheses:

A variation is to not allow two digits to be placed together to make a two digit number (done above only for some of the numbers 29 or larger); here are some: