What is the best and most up-to date Ancient Greek-English dictionary?

Depends on your criteria.

Biggest & Up to date is not English, but the now online DGE Diccionario Griego-Español . Only goes up to epsilon though, and I don’t see it finishing for another century.

Biggest in English remains Henry George Liddell, Robert Scott, A Greek-English Lexicon — though the online editions don’t include the 1996 Supplement.

The Cambridge Greek Lexicon is coming out next year; it’s not meant to be as big as LSJ, but it has been redone from scratch, rather than copypasting previous lexica (a tradition LSJ itself is part of).

The Brill Dictionary of Ancient Greek came out last year, as a translation of Montanari’s Italian dictionary. I haven’t gone through it; from the headword count, it sounds close to LSJ (more than the original edition, less than original + supplement), and I know that Montanari maintained PAWAG-Poorly attested words in ancient greek, with 1000 words not in LSJ (well, a substantial subset of them, anyway).

It won’t be as comprehensive as DGE, which quite confidently does Proper Names and Early Byzantine texts, an area previous dictionaries have shied away from. But then again, DGE is up to epsilon.