Quick answer
a lo mejor
Natural: maybe / perhaps
Literal: “at the best”
How it is used
Where it is said
Very common in Spain and widely understood elsewhere; in much of Latin America <b>capaz que</b> or <b>tal vez</b> is heard more often in the same slot.
A mini dialogue
Watch out
a lo mejor
A lo mejor always takes the indicative: a lo mejor viene, never a lo mejor venga. This is the difference from quizás and tal vez, which can take the subjunctive (quizás venga) to sound less certain.
a lo mejor
It has nothing to do with quality. It does not mean “at best”; that is como mucho or en el mejor de los casos.
Check yourself
Which sentence is correct?