Here's a random house rule I just thought up for Lamentations of the Flame Princess. It might work, or it might be complete crap - I don't know. I might as well see what the internet thinks.
You know how in some versions of D&D and related games, fighters can have certain abilities like attacking all enemies within melee range if they're under a certain number of hit dice, or attacking a second enemy on their turn if their first attack kills an enemy ("Cleave" in 3E parlance), or making multiple attacks per combat round once they reach certain level thresholds? In the Rules As Written, fighters do not have these kinds of multiple attack features. This was a deliberate design choice on the part of the game designer, of course, but some people might want to house-rule some kind of multi-attack feature into the game.
Now, you know how sometimes players ask if they can try to pull off various oddball stunts that either aren't covered or aren't generally allowed within the rules? "Can I time my shot so that it hits both guards when they're lined up?" "Can I boost off the wall and spin kick the enemy for extra damage?" "Can I feint to the right before attacking from the left to throw him off guard?" "Can I drop on top of him from the ledge with my sword pointed down and stab him extra hard and maybe smash him because of the gravity?" "Can I grab his club out of his hand when he swings at me?" As far as I can tell, whether or not these kind of actions are allowed is meant to be decided by the referee, as is the way of adjudicating such actions if they are allowed. Someone going strictly by the law of R.A.W. would probably disallow them, as they aren't covered by the rules, and the rules in D&D and OSR games tend to be stricter and more codified in combat than outside of it, for a variety of reasons. Since I would bet that most referees don't play strictly in the R.A.W., I would imagine that a lot of them (myself included) would generally allow these kinds of stunts or ones like them in certain conditions.
Both of these types of combat actions - multiple attacks and complicated/unusual/acrobatic stunts - generally seem to me like things that an experienced combatant could probably pull off with more consistency than an inexperienced combatant. They also seem like the kinds of things that someone specializing in combat - a member of the "fighter" class - would be vastly more likely to achieve than others. So let's base this house rule on level and class.
The rule: If a fighter attempts a combat maneuver (including but not limited to making multiple attacks in one combat round, hitting multiple characters with one attack, or moving and attacking at the same time) which is not covered by the rules or generally not permitted by the rules, and the referee permits the fighter a chance at success, the fighter rolls a d20 and adds their own level. If the sum is 21 or higher, the fighter can attempt the maneuver. Fighters of level 20 and higher still only add 19 to this roll, so that a roll of 1 is always a failure. Non-fighter PCs can also attempt such combat maneuvers, but they are always treated as level 1 fighters for such purposes. If any other roll (to-hit, saving throw, etc.) would normally be needed for success (e.g. with multiple attacks), the fighter must still make that roll as well.
TL;DR If the fighter wants to do some crazy move in battle, they roll a d20 and add their level. 21 or better to succeed. Caps at level 19. Referee might make you roll some other dice, too. Other classes can do it, but only as a level 1 fighter.
Alternatively, you could probably come up with something that uses the character's base attack bonus (which only improves beyond +1 for fighters in LotFP, making it a de-facto fighter class ability), or that uses some kind of saving throw. You could also try to come up with a formula in which the ability maxes out much sooner than the nigh-mythical level 19. Hence that whole "it might be complete crap" thing I typed earlier; this isn't a very elegant house rule as I've written it.
You could also codify what kinds of things are or are not permitted as "combat maneuvers" in the game, but to some extent this violates the spirit of this house rule. The reason I came up with it was to make a simple and unified way of adjudicating whatever crazy schemes the players come up with, and thus allow or encourage creative, OSR-style shenanigans that inflexible board game-style or video game-style rules couldn't cover due to the sheer number and variety of possibilities. You could easily modify this rule to give magic-users a level-dependent chance to spontaneously alter their spells, or to give clerics a level-depended chance to successfully pray for divine intervention.
In a game that already does allow multiple attacks or cleaving or whatever (i.e. not LotFP), you could still probably use this house rule for other weird things that players try, like suplexing a naga onto a bed of nails (which might also require a grapple check at the very least) or screaming directly into a morlock's ear while gut-punching him or catching the giant turtle's shell just right with your sword to make it do some sweet flips.
I don't think this house rule is covering new ground at all. I've heard of similar ideas before. Honestly, this is probably Baby's First Game Design-level stuff, and most D&D/OSR referees are already way ahead of me on this count. Mostly, I'm just looking for input on whether or not this might be a good way to give fighters more mechanical options and creativity in combat, and what some better methods of achieving this might be. If the fighter says "I run up the wall and back-flip, then slice the monster on the way down," and for whatever reason you want to allow it, what comes next? As the referee, how do you determine whether or not the fighter succeeds, and how well?