The cuneiform tablet MS 2351 from the 19th century BC contains the 15-digit sexagesimal number 13 22 50 54 59 09 29 58 26 43 17 31 51 06 40, which happens to equal $20^{20}$. I also seem to remember they constructed a table of reciprocals for numbers of the form $125 \cdot 2^n$ for exponents up to $19$.
Edit. Last year some fragments of a large table have been identified: Ossendrijver discovered that the complete table contained the powers of 9 up to $9^{46}$.
