Depends on how much time is available to buff, whether you're considering buffs to others or just yourself, and to what extent limited-use abilities are considered - a magus can nova very nicely for example. Also on at what level you're looking at.
Inquisitors get judgement, bane and divine power/favour
Which isn't bad.
That doesn't answer the other questions though. A bard can add a lot of damage if you're including buffs to others, a magus can deliver a big smackdown a few times a day, a fighter can keep doing good damage forever so long as the healing holds out, given a little time to buff an inquisitor or warpriest can be an engine of doom - you need to define some limits on the scenario to get a meaningful answer.
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The Inquisitor's Destruction Judgement, Ranger's Favored Enemy, sneak attack dice. For the Fighter, weapon spec (bonus feats are a class feature). All of these grow over time, different rates obviously. These are the low hanging fruit. If all you want is more damage per round, I'm sure that has been discussed on the forums before (probably never stopped being discussed actually).
It depends on how you define best. Some classes have very circumstantial bonus, and or limited uses of class abilities. Others have constant abilities. Another thing to keep in mind is that often the best boost to damage is to boost your chance to hit. If you miss it does not matter how high your damage is.
Paladins get smite evil which is probably one of the best extra damage class abilities in the game when in play. But it is both limited in scope and has very few uses per day. Against a single powerful evil target it is hard to beat especially if the target is an evil outsider, dragon or undead. Not only does it add level to damage (x2 level to damage on the first hit vs evil outsiders, dragon or undead), but it also adds your CHA to hit and AC, it also ignores all damage reduction. But if the target is not evil it is completely useless. It also does not work well vs a horde of lesser evils because the paladin simply does not have enough of them.
And inquisitor has multiple stacking abilities that if he manages to active all do quite well. The two problems are having enough time to get everything going, and all his abilities have limits on how often they can be used. If he manages to get everything going he comes close to the paladin, but has the advantage of being able to tailor the effect to any type of creature.
Rangers favored enemy can be a huge boost if you will be facing that type of enemy. Unlike most damage boosting class abilities they have no limits on how often they can use it. It increases both the damage and the chance to hit. It works best with a fighting style that gives more attacks per round, like archery or two weapon fighting. Unlike most class abilities it works extremely well vs fighting a large number of less powerful foes.
Fighters have the most consistent damage boost of any class. Combining fighter only feats like weapon specialization and greater weapon focus with weapon training gives fighters a boost in damage vs anything they fight against. They also have enough feats that they can afford to pick up feats that other classes ignore. Most people consider improve critical to be a waste of a feat, but a fighter can easily pick it up without having to worry about wasting a feat.
This is not a comprehensive list and I am sure there are things I left off.
barb - rage +2/3 damage, furious weapon +1
Ubarb - rage +2 damage and the damage style for +1 +1 per four levels, furious weapon +1
bard - inspire courage maybe count good hope +2 maybe arcane strike +1/2/3
bard dawnflower dervish - inspire x2 maybe count good hope +2 maybe arcane strike +1/2/3
cleric - spells and a few domains that'll add 1/2 level to damage for an attack and/or rage for +2/3.
Druid - wildshape for str and extra attacks if those count.
fighter - going to count fighter feats into this - +2 at lv4, +1 at 5 and nine, +2 via gloves of dueling.
fighter mutation warrior - mutagen +2/3
paladin - smite level to damage, sacred bond weapon for +1 at 5 and another +1 at 9.
Ranger - +2/4/6 to FE
rogue - ~3.5 per qualifying hit every odd level.
Inquisitor - judgement +1 and +1 per 3. Bane +2 and ~7. and domain.
Alchemist - mutagen +2/3, maybe extra natural attacks
Cavalier - challenge for level to damage, order for +1 and +1 per 4 potentially mounted charges for x2/3 damage
cavalier - order of pike same as above but with weapon training for +1 at 5 and another +1 at 9, gloves of dueling +2.
magus - arcane pool +1 and +1 per 4 after that. access to fighter feat +2, maybe arcane strike.
magus Myrmidarch - access to weapon training at reduced rate +1 and gloves of dueling +2.
Hunter - animal focus for str, doesn't stack with str belt.
investigator - mutagen +2/3 and studied combat +1/2 level, combat inspiration with inspired weapon addes 2x inspiration to damage.
Brawler mutagenic mauler - mutagen +2, at lv6 +2 damage.
bloodrager - rage +2/3, furious weapon +1, maybe arcane strike
shaman - one spirit gives bane eventually.
skald - raging song, maybe good hope, maybe arcane strike
slayer - +1/2/3 studied target and sneak attack ~3.5 per 3 on qualifying.
swashbuckler - level to damage, double it on one attack a round, fighter bonus feats +2
warpriest - fervor divine favor +1/2/3/4 (FF), sacred weapon vicious ~7 and +1 at 8, fighter feats +2
warpriest arsenal chaplain - same + weapon training at 5 +1 and 9 +1 with gloves of dueling +2
medium - taboo +2, spirit bonus +1 and +1 per four, with a feat for +1.
medium halfling - same but +1 per 3.
occultist - resonance power str, doesn't stack with belt. Legacy weapon bane +2 +~7 and another +1 at 6
That should be most the classes that have damage boosts worth mentioning.
Druid - nature fang archetype w/ crocodile domain. Studied Target, Sneak attack, Ranger Fighting Styles, druid buffs & summons
