Green’s Dictionary of Slang

(orig. US) one’s very best efforts.

1869		Hartford (CT) Courant 4 Oct. n.p.: How comes it that the friends of F.A. Marcy did their level best [...] to secure the defeat of Mr. Francy?
1874		‘English Sl.’ in Eve. Telegram (N.Y.) 9 Dec. 1/5: Let us present a few specimens:– [...] ‘He did his level best.’.
1883		Bulletin (Sydney) 6 Jan. 5/1: The Mayor is doing his level best to hurry on the trial; but […] the hair that suspends Damocles’ sword is as new and as strong as ever.
1892		H. Nisbet Bushranger’s Sweetheart 106: I have done my level best to cater for them.
1899		C. Rook Hooligan Nights 150: Never you mind, cocky, you done yer level.
1901		Boy’s Own Paper 6 Apr. 421: I [...] if I do say it myself, had put my level best into my work.
1917		‘A-No. 1’ From Coast to Coast with Jack London 130: I tried my level-best to dissuade him from this notion.