Wiktionary
n. (context idiomatic English) The very best that one can do
WordNet
Wikipedia
Level Best is a compilation of the largest successes of the UK band Level 42. It was released at the end of 1989, marking a decade since the group's beginnings.
Usage examples of "level best".
The computers have identified him as a bunch of pixels that require a great deal of attention and bandwidth, and so somewhere some poor algorithm is churning through the high-pressure slurry of compressed pixels that is the image of Andrew Loeb, and doing its level best to freeze the most rapidly-moving parts into discrete frames and chop them up into checkerboard-squares that can be broadcast as packets over the Net.
As an outsider, you do your level best, but you never really know.
And in _Worlds of Wonder_, Silverberg does his level best lo convey the basic mechanics.