Monks get goofy "inner mind" type stuff, Paladins and Rangers get spellcasting from the Gods, and Barbarians rage.
Since Fighters are about fighting and mastering certain weapon styles, perhaps they can gain stuff related to fighting styles. At middle to high levels, when you're fighting giants and dragons and archmages, the Fighter can replicate certain magic spells and do superhuman actions. A nimble Fighter can ignore negative effects of terrain, act hasted through an "adrenaline surge," and jump really high/glide. An inspring figure can give morale bonuses like a Bard, negate mind-affecting spells on himself and others, and stave off damage and grant hit points "through second wind/fighting through the pain." For things that Clerics can do (summon monsters, banish foes), a Fighter might get duplicate abilities ("the cavalry has arrived"/rescue attempt of minions like summoned monsters, staring down an opponent and getting him to surrender/relent like save-or-suck spells).