Four Fours/Table of 4s
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Believe it or not, the rules allow you to do quite a bit with a single 4. The rules say you may combine 4s with any mathematical symbol except numbers. Thus, in addition to 4 alone, we also have:
The following fractions are also useful:
Could possibly add constants (harmonic number), possibly special functions.
Once you allow variables like x into the mix, it's lights out.
One 4 with Variables
These lists blow up pretty fast... as you can see, focusing on using a smaller number of 4s can force you to be creative. This makes it possible to combine 4 4's beyond the integers from 1 to 20, and keep on going...
where is any number expressible with two 4's.
(That is, 74 is 4A in hexidecimal, or base 16 = 4 * 4)