I don't think there is a true equivalent, due to the nature of effort.
"little to no effort" is common because of the potentially for pedantry when it comes to claiming no effort whatsoever. Consider the exchange where a teacher says a child put in no effort, and the child counters with the fact of them writing their name proving some (if negligable) effort.
On the other hand, maximum effort is poorly defined. While no effort means doing absolutely nothing towards the task at hand, putting in total effort is hard to pin down, and the difference is far less likely to be argued. Even terms for maximum possible effort are relatively unusual (maximum and total both sound odd). There are a few phrases that are more common such as "giving 100%/evertything/it all/all you've got" or "doing your best". Terms for almost maximum to be equivalent to the little are even less prevelant.
The closest I can get while sounding natural would be:
giving everything (or as close as makes no difference)
Generally, one would just use a term for a lot of effort, or one of the 'maximum effort' terms suggested above. I can't think of a situation where it's necessary or natural to specific close to maximum effort.
Edit: The same is essentially true of most things. Let's take betting, betting almost everything to everything still sounds pretty awkward. If we reuse:
Despite his confidence, he bet little to nothing on the outcome of the game.
Because of his confidence, he went all in (or as close as makes no difference) on the outcome of the game.
I can't think of anything better generically. If you are willing to stray a little 'effectively' might work, E.g: 'effectively going all in'. The effectively has the meaning of close enough to be the same, which gives mostly the same impression.