From Number Puzzles to Automata

There’s an old puzzle that requires the solver to form as many whole numbers as possible, starting with 1, using only the digit 4 exactly four times, together with ordinary arithmetic and algebraic notation.

How far you can get depends on what you mean by “ordinary notation.” Using just decimals, parentheses, concatenation (44), and the symbols for addition, subtraction, multiplication, and division, you can readily reach 20.