The Vilna Gaon explains that there were three things Achashverosh performed for Haman; he made him great by making him financially wealthy, he elevated his by giving him more authority, and he gave his a higher seat in that his advice had superiority over that of Achashverosh’s other advisers.
The Yalkut Shimoni (1053) and M’nos HaLevi say that these three actions indicate that Achashverosh gradually gave Haman new powers every day. The Yalkut Shimoni actually goes on to say that Haman eventually had even more power than the king, Achashverosh building for him a throne even higher than his own. Rav Avie Gold points out that the gematria of Haman (5+40+50=95) is equal to that of “hamelech,” (“the king”) (5+40+30+20=95), implying that they became equal to each other. Furthermore, the gematria of Haman’s entire given name here, Haman son of Hemdasa the Aggagite, (5+40+50+2+50+5+40+ 4+400+1+5+1+3+3+10=619) is one less than “kesser,” (“crown”) (20+400+200), implying that he was just within reach of the royalty he desired.
The Malbim points out that Haman’s gradual ascent to power is similar to what Yosef tells his brothers (Bereishis 45:8) regarding the three steps of his promotion. The Alshich explains the reason for this (in both cases, perhaps) is that king did not want his others advisers to harbor jealousy toward the new “upstart.”
Perhaps another reason for this gradual elevation is that Achashverosh, himself, was not a product of royalty, but was a self-made man, and felt that this was the proper way for somebody to advance.
The Vilna Gaon points out that this reference to the “king,” is, again, H-Shem. In His Wisdom, H-Shem elevated this wicked man in order to provide the platform for the Purim miracle.