The first definition of the powers beyond the third is probably in Diophantus' *Arithmetic* (written somewhere from 100 BC to 300 AD). 

In the introduction to that work, he defines fourth powers ("dynamodynameis", lit. "square-squares"), fifth powers ("dynamokuboi", lit. "square-cubes"), and sixth powers ("kubokuboi", lit. "cube-cubes"). Throughout the course of the work, he uses them a lot. Also, he assigns them no geometrical significance whatsoever, just as he doesn't think of squares and cubes themselves in a geometric way; e.g., he sees no problem in adding the square of the unknown to a constant, etc.