According to this page the earliest known use of integers to represent repeated multiplication is by Nicole Oresme in the mid 1300s. However, he didn't use a raised integer notation. The rest of this answer is taken from that page.
Nicolas Chuquet used raised integers in 1484, though for him $12^3$ was a shorthand for $12x^3$.
In 1636 James Hume used roman numerals as exponents, e.g. for $12^3$ he would have written $12^\textrm{iii}$, but apart from that minor distinction he was essentially using modern notation.
Rene Descartes used raised arabic numericals as exponents in 1637, with the exception that he tended to write $xx$ rather than $x^2$, though he would still write $x^3$, $x^4$ etc. He wrote:
...$aa$ ou $a^2$ pour multiplier à par soiméme; et $a^3$ pour le multiplier encore une fois par $a$, et ainsi à l'infini.
which roughly translates as
...$aa$ or $a^2$ to multiply by itself, and $a^3$ to multiply again by $a$, and so ad infinitum.