The cuneiform tablet <a href="http://www.schoyencollection.com/mathematics-collection/arithmetics/ms-2351">MS 2351</a> 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: 
<a href="https://arxiv.org/ftp/arxiv/papers/1306/1306.5989.pdf">Ossendrijver</a> discovered that the complete table contained the powers of 9 up to $9^{46}$.