Agmondesham Vesey (1708 – 3 June 1785) was an Irish politician and the second husband of Elizabeth Vesey, one of the founders of the Blue Stockings Society.
He was Member of the Parliament of Ireland for Harristown, County Kildare, and Kinsale, County Cork, who held the appointment of accountant-general of Ireland, probably from 1767.
In 1746 he married Elizabeth Vesey, a cousin of his. The couple had no children together, and Agmondesham was continuously unfaithful to Elizabeth but she maintained the façade of a happy marriage. Elizabeth nursed her husband through attacks of epilepsy, but depended for her support upon a circle of female friends. The couple split their time between London, England and Lucan in Ireland, but eventually settled mostly in London at houses in Clarges Street and Bolton Row, Mayfair, where Elizabeth hosted her intellectual salon parties. where entertainment consisted of conversations on literary subjects.
Agmondesham Vesey died on 3 June 1785, leaving his wife and her companion, Miss Handcock, facing relative poverty, Agmondesham having left them nothing in his will, despite leaving £1000 to his mistress.