What are the abilities that boost moves and what are their multipliers?

Question

ex: iron fist x1.3 to punching moves

Comments

gen 6 is here so there might be some additions...

Answer 1

This is the complete list up to Generation 9.

Boosts Base Power of Move:

Aerilate: Turns Normal-type moves into Flying-type and boosts the Base Power by 20%
Aura Break: If Dark Aura or Fairy Aura is present, decrease the Base Power of all Dark- or Fairy-type moves by 25%, respectively
Battery: Provides 30% boost to ally's special moves
Blaze: Provides 50% boost to Fire-type moves if user's HP is below 33%
Dark Aura: Provides 33% boost to all Dark-type moves
Dragon's Maw: Provides 50% boost for Dragon-type moves
Electromorphosis: Provides 100% boost to next move, assuming it is a damaging Electric-type move, after being hit by an attack
Fairy Aura: Provides 33% boost to all Fairy-type moves
Flare Boost: Provides 50% boost to special moves if user is burned
Galvanize: Turns Normal-type moves into Electric-type and boosts the Base Power by 20%
Iron Fist: Provides 20% boost to punching moves
Mega Launcher: Provides 50% boost to pulse/aura moves (Heal Pulse heals 100% of target's HP instead of 50%)
Normalize: Turns all moves into Normal-type moves and boosts all attacking moves by 20%
Overgrow: Provides 50% boost to Grass-type moves if user's HP is below 33%
Pixilate: Turns Normal-type moves into Fairy-type and boosts the Base Power by 20%
Power Spot: Provides 30% boost to ally's physical moves
Punk Rock: Provides 30% boost to sound-based moves; takes half damage from opposing sound-based moves
Reckless: Provides 20% boost to moves with recoil, excluding Struggle
Refrigerate: Turns Normal-type moves into Ice-type and boosts the Base Power by 20%
Rivalry: Provides 25% boost to moves used against a Pokemon of the same gender, but decreases power by 75% if used against the opposite gender.
Sand Force: Provides 30% boost to Ground/Rock/Steel-type moves in Sandstorm; user is immune to Sandstorm
Sharpness: Provides 50% boost to slicing moves
Sheer Force: Provides 30% boost to moves with a beneficial secondary effect, but removes that secondary effect
Steely Spirit: Provides 50% boost to user's and its ally's Steel-type moves
Strong Jaw: Provides 50% to biting moves, excluding Bug Bite
Supreme Overlord: Provides 10% boost for each ally fainted, up to 5 allies (so maximum 50% boost)
Swarm: Provides 50% boost to Bug-type moves if user's HP is below 33%
Technician: Provides 50% boost to moves with 60 Base Power or under
Torrent: Provides 50% boost to Water-type moves if user's HP is below 33%
Tough Claws: Provides 30% boost to contact moves
Toxic Boost: Provides 50% boost to physical moves if user is poisoned
Transistor: Provides 30% boost to Electric-type moves
Water Bubble: Provides 100% boost to Water-type attacks; user takes halved damage from Fire-type moves; user is immune to burn
Wind Power: Provides 100% boost to next move, assuming it is a damaging Electric-type move, after being hit by an wind move

Boosts Offensive Stats:

Defeatist (technically not a boost but still a multiplier): Decreases offensive stats by 50% if user's HP is below 50%
Flower Gift: Provides 50% boost to user's and its ally's Attack and Special Defense
Gorilla Tactics: Provides 50% boost to Attack stat, but can only use the first move it selects after switching in
Guts: Provides 50% boost to Attack if there is a status condition
Hadron Engine: Provides 33% boost to Special Attack and sets up Electric Terrain
Huge Power: Provides 100% boost to Attack
Hustle: Provides 50% boost to Attack, but lowers accuracy by 20%
Minus: Provides 50% boost to Special Attack if ally has Plus or Minus
Orichalcum Pulse: Provides 33% boost to Attack and sets up harsh sunlight
Plus: Provides 50% boost to Special Attack if ally has Plus or Minus
Pure Power: Provides 100% boost to Attack
Rocky Payload: Provides 50% boost to offensive stat when using a Rock-type move
Slow Start (also not a boost): Decreases Attack and Speed by 50% for the first 5 turns the user is on the field
Solar Power: In harsh sunlight, provides 50% boost to Special Attack, but user loses 12.5% health every turn
Steelworker: Provides 50% boost to offensive stat when using a Steel-type move

Provide Multipliers to Other Boosts/Provides Other Boosts:

Adaptability: Turns STAB boost from 50% to 100%, effectively powering up STAB moves by 33%
Analytic: Provides 33% boost if user moves last (this includes the opponent switching)
Flash Fire: Provifes 50% boost to Fire-type moves after being hit by a Fire-type attack (loses this boost after switching out); user is immune to Fire-type moves
Merciless: All moves are critical hits (50% boost) if target is poisoned
Neuroforce: Provides 20% boost if the move is Super Effective
Parental Bond: Attacks hit twice, with second attack dealing 25% of the damage (effective power increase by 25%)
Sniper: Provides 50% boost to moves that result in a critical hit
Stakeout: Provides 100% boost to moves if the target switches in that turn
Tinted Lens: Provides 100% boost to moves that are not very effective

Answer 2

Analytic- raises the power of moves by 30% if the user moves last.

Blaze- raises fire attacks to 1.5 times power when HP is below 1/3.

Flash Fire- raises fire attacks to 1.5 times power when hit by a fire type attack, also grants immunity to fire type attacks.

Guts- Raises attack stat to 1.5 times regular damage when a status ailment is in effect, burn still lowers attack.

Hustle- Physical attacks do 1.5 times regualr damage, but average accuracy is only 80%.

Minus- Raises Sp. attack to 1.5 times damage when battling with a pokemon that has Plus.
Plus- Raises Sp. attack to 1.5 times damage when battling with a pokemon that has Minus.

Overgrow- raises grass type attacks to 1.5 times damage when HP is below 1/3.

Prankster- makes status moves have a priority of +1.

Reckless- increases recoil moves power by 20%.

Rivalry- raises base power of moves by 25% if the opponent is of the same gender, but decreases the base power of moves by 25% if the opponent is of the opposite gender.

Sand force- raises the power of Rock, Steel, and Ground type attacks by 30% in a sandstorm.

Sheer force- raises the base power of all attacks by 30% but ignores any secondary effects.

Swarm- raises bug type attacks to 1.5 times power when HP is below 1/3.

Technician- raises the power of moves to 1.5 times regular power, but only if the move has a base damage of 60 or less.

Torrent- Raises the power of water type attacks to 1.5 times power when HP is below 1/3.

Toxic boost- raises the user's attack by 50% if poisoned.

I might have missed one or two but these were the ones I could list. Hope it helps!