Belly Drum + Rest
Strategy: Use Belly Drum to max out your attack, then use Rest so that your HP becomes full once again. Works better with a Chesto Berry.
Belly Drum + Psych Up
Strategy: 2v2 only. One Pokemon uses Belly Drum, and the other one uses Psych Up to copy the max attack stat change.
Charge + Electrical Attack
Strategy: Charge, then use an Electrical Attack for double damage.
Defense Curl + Rollout
Strategy: Defense Curl first, then Rollout for x2 damage.
Dragon Dance + Physical Attack
Strategy: Dragon Dance as much as you'd like, then use the Physical Attack of your choice for powerful sweeping.
Endure + Reversal/Flail
Strategy: When you believe that your opponent is about to strike a final blow on you, use Endure so you'll survive with 1 HP. Then use Reversal/Flail for heavy hits. Works better on faster Pokemon.
Flatter + Psych Up
Strategy: Use Flatter, then Psych up to copy the upped special attack stat change and not the confusion status.
Foresight + Fighting Move
Strategy: Versus a ghost type Pokemon, use Foresight first, then use any Fighting Move you want to and it will hit for damage as it goes around the ghost type advantage completely.
Leech Seed + Ingrain
Strategy: Leech Seed first for damage, then Ingrain for lots of HP gain. Leftovers as a held item makes this strategy work even better.
Lock On + Zap Cannon
Strategy: Using Lock On will ensure that Zap Cannon hits, which has a 50% chance of hitting normally. Zap Cannon also gives an added paralyzing bonus.
Mean Look + Perish Song
Strategy: Mean Look first, then use Perish Song. If you can survive until the turn before both Pokemon faint, then switch.
Mimic + Imprision
Strategy: Mimic whichever move you want to ban the other pokemon from using, then Imprison it.
Rain Dance + Water Attack
Strategy: Rain Dance first, then use any water attack for 1.5x damage.
Rain Dance + Thunder
Strategy: Rain Dance first, then use Thunder for 100% accuracy.
Rest + Sleep Talk
Strategy: Rest first, then Sleep Talk. A not-so-great strategical combo because it's random.
Rest + Snore
Strategy: Rest first, then Snore for damage. Not much to say here.
Safeguard + Thrash/Outrage/Petal Dance
Strategy: Using Safeguard first, use Thrash/Outrage/Petal Dance. At the end of those attacks, you will not get confused because of the Safeguard.
Spore + Focus Punch
Strategy: Spore, then Focus Punch. Relies a little bit on luck seeing as your opponent might wake up and attack you to disrupt Focus Punch.
Subsitute + Flail/Reversal
Strategy: Subsitute 4 times with your opponent knocking out each of them, then use Flail/Reversal for big damage. Works better on faster Pokemon.
Substitute + Focus Punch
Strategy: Subsitute first, then next turn your opposing pokemon won't be able to directly hit you. That means that you can pull off a Focus Punch even if your opponent attacks you and hits, because the subsitute will take the hit. This works best if your Pokemon is slower than the opponents.
Sunny Day + Solar Beam
Strategy: Sunny Day first, then Solarbeam as much as you'd like with no startup phase.
Sunny Day + Synthesis/Morning Sun/Moonlight
Strategy: Use Sunny Day, then Synthesis/Morning Sun/Moonlight to restore all HP.
Sunny Day + Fire Attack
Strategy: Sunny Day, then use any Fire attack for 1.5x damage.
Stat Up Move + Baton Pass
Strategy: Use any move to increase your pokemon's status, then use Baton Pass to give those changes to any other Pokemon of your choice.
Swagger + Psych Up
Strategy: Use Swagger to confuse your opponent, then use Psych Up to copy the upped attack but not the confusion.
Swagger + Screech
Strategy: Use Swagger, then use Screech to move the defense of the opponent down as much as you'd like to greatly increase the damage taken when they hit themselves in their confusion. Works well in conjunction with attacking moves as well.
Toxic + Dig/Fly/Dive/Bounce
Strategy: Use Toxic, then use Fly continuously to make opponents hit half of the time. Doesn't work well if your opponent has something to hit you in the air/ground though.
Toxic + Protect/Detect
Strategy: Toxic first, then just keep using Protect or Detect. Not much else to say here.