Drow + Visage = very powerful familiars. Add Venge or SFiend for additional pain.
Drow + Windrunner = very high-damage Focus Fire (add Desolator on WR for tower melting)
Drow Precision Aura (or Lycan Howl) + Clinkz / Windrunner = very effective Strafe / Focus Fire
Spectre + Troll = very fast-hitting Haunt illusions (with max Desolate). Hell on supports. Requires dual lanes, or for Troll to go mid (preferably against a weak enemy mid). You might get away with "support Troll" if no aggro-tri against you.
Faceless Void + Witch Doctor / Jakiro / Lich = very good damage-dealing synergy with Chrono, plus defensive supports to protect a farming 1-position Void. Note: it's far easier to ban out the Void than trying to ban out the multiple supports that are good with Chrono!
Wisp + Tiny/CK/Sven + Nature's Prophet = a very powerful global ganking combination. While you may not be able to prevent the Wisp + Carry, at least don't let them get both Wisp and NP! AA, Clockwerk, and Invoker are also good with this combo.
Lycan + Death Prophet = painful pushing power. Lycan runs around the back to demolish supports, while Exorcism goes to work on the rest of the team.
Lycan + Broodmother = Howl makes for very dangerous broodlings.
Mirana + Bane / Shadow Demon / Crystal Maiden = very good set-up for arrows.
Visage = heralds an aggro tri-lane (usually). May also prompt a Drow last-pick.
Vengeful Spirit = heralds a roaming support or roaming pair. Avoid leaving squishy carries alone if they don't have an escape skill!
Earth Spirit = heralds a WTF (just kidding!)
Disruptor + Axe / Undying = very mean "cage match" with Helix or Tombstone. Sometimes done as an offlane duo, or as part of aggro tri-lane. Jakiro also combos really well with this. Countered by heroes with escapes or healing supports.
Centaur / Shadow Shaman + Skywrath Mage = good control and good damage; very potent roaming pair
Clockwerk + Skywrath = put enemies in a box and nuke them, while interrupting them constantly.
Clockwerk + Invoker (Exhort) = Cogs makes the killbox, Sunstrike makes the kill.
Nature's Prophet + Skywrath = surround them with trees and nuke them from the sky.
Magnus + Anti-Mage/Phantom Assassin/Faceless Void = carry doesn't need Battlefury to farm effectively. Hard carry can do massive damage during Reverse Polarity.
Warlock + Undying/Elder Titan/Kunkka/Phoenix = it's bad when you're in a massive pool of syrup and can't get out of the way. Add Pugna with Nether Ward for additional enemy frustration. It's a good idea to pick another support with a teamfight ult so that the enemy has to choose between targets to interrupt. See also Team Ideas around Delayed AOE.
Anti-mage or Phantom Assassin = may prompt a last-pick Magnus, for the "free Battlefury" from Empower.
Heroes that are Helped by Heroes:Anti-mage --> helped by --> Magnus (Empower = "free Battlefury")
Axe --> Disruptor, Undying
Bloodseeker --> Zeus (get targets low health faster)
Bounty Hunter --> Lycan (Howl), for massive Jinada strike
Clockwerk --> Invoker (Sunstrike), Skywrath (ult), Gyrocopter (Call Down), Lina/Lesh, Kunkka (Ghost Ship)
Doom --> Abaddon, Dark Seer, Dazzle
Dragon Knight --> Beastmaster aura, Crystal Maiden aura (DK needs AS and mana regen)
Faceless Void --> Dark Seer, Magnus (setup for Chrono)
Undying --> Broodmother, Nature's Prophet, Chen (numbers around Undying boosts Soul Rip)
See also: List of Initiators by Ability Type