A simple problem came to mind a while ago, but I couldn't think of a clear solution.
We can name a function for this as maxlib(n).

A brute force approach to compute maxlib has insane time complexity. I actually wrote and ran a program to compute this brute force, and could only reach n = 6 after about 2 hours, maybe.

I posted a programming challenge about this, and still haven't got a fast enough solution, but a user named Loopy Walt did post a convincing conjecture that,
This conjecture is justified although not proved by the following group patterns.
An OEIS sequence (A320666) exists for maxlib. The currently known values are only up to n = 24.

I can't understand the programming aspects but I can glean that obviously since a 1x1 board would give zero liberties there's a non-linear component to it. However wouldn't a group with the maximum liberties be, for example on 19x19, a 63-stone near loop? So wouldn't it just be (4*(2n-2)) minus whatever constants to account for 1x1?

As said in the OP, this group has 229 liberties.

Click Here To Show Diagram Code

[go]$$
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | X X X X X X X X X X X X X X X X X X X |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . , X . . X . , X . . X . , X . . |
$$ | . X . . X . . X . . X . . X . . X X X |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . . X . . X . . X . . X . . X X X |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . , X . . X . , X . . X . , X . . |
$$ | . X . . X . . X . . X . . X . . X X X |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . . X . . X . . X . . X . . X X X |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . , X . . X . , X . . X . , X . . |
$$ | . X . . X . . X . . X . . X . . X X X |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ ---------------------------------------[/go]

Click Here To Show Diagram Code

[go]$$B
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | X X X X X X X X X X X X X X X X X X X |
$$ | . X . . . . . . . . . . . . . . . . . |
$$ | . X . , . . . . . , . . . . . , . . . |
$$ | . X X X X X X X X X X X X X X X X X X |
$$ | . X . . X . . . . . . . . . . . . . . |
$$ | . X . . X . . . . . . . . . . . . . . |
$$ | . X . . X X X X X X X X X X X X X X X |
$$ | . X . . X . . X . . . . . . . . . . . |
$$ | . X . , X . . X . , . . . . . , . . . |
$$ | . X . . X . . X X X X X X X X X X X X |
$$ | . X . . X . . X . . X . . . . . . . . |
$$ | . X . . X . . X . . X . . . . . . . . |
$$ | . X . . X . . X . . X X X X X X X X X |
$$ | . X . . X . . X . . X . . X . . . . . |
$$ | . X . , X . . X . , X . . X . , . . . |
$$ | . X . . X . . X . . X . . X X X X X X |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ | . X . . X . . X . . X . . X . . X . . |
$$ +---------------------------------------+[/go]

This arrangement of the stones is probably preferable for systematic reasons.

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The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)