In sense “upstart person”, applied to 15th century William de la Pole, 1st Duke of Suffolk, one of the first nouveau riche nobles (risen from merchant class).
Later mis-analyzed as Jack-an-apes (16th and 17th century), leading to folk etymology (taking “ape” from “monkey”).
Also attributed to the transition of the playing card 'Knave' to 'Jack' where both cards were associated with the idea of roguery. The 'Jack' became the Jack a napes, derived from Jack a naipes, naipes being the Spanish for playing card.

jackanapes (plural jackanapeses)

1. (obsolete) A monkey.
2. (dated, pejorative) An impudent or mischievous person.
   • 1611 April (first recorded performance)​, William Shakespeare, “The Tragedie of Cymbeline”, in Mr. William Shakespeares Comedies, Histories, & Tragedies: Published According to the True Originall Copies (First Folio), London: Printed by Isaac Iaggard, and Ed[ward] Blount, published 1623, OCLC 606515358 ↗, [Act II, scene i], pages 375–376 ↗:

     Was there euer man had ſuch lucke? when I kiſt the Iacke vpon an vp-caſt, to be hit away? I had a hundred pound on't: and then a whoreſon Iacke-an-apes muſt take me vp for ſwearing, as if I borrowed mine oathes of him, and might not ſpend them at my pleaſure.

1. plural form of jackanape