I had to edit this after someone pointed out that Hidden Abilities aren't allowed.
Parents: Quagsire & Dunsparce
Shared egg group: Field
Offspring name: Dunsire
New type: Normal/Ground
New base stats: 108/88/88/75/75/50
New ability and desc: Serene Grace + Water Absorb = Oceanic Serenity: Chances of move's effects are doubled. If hit by a Water-type move, the next move used has a guaranteed effect. Immunity to Water.
Notable moves: Coil, Stealth Rock, Body Slam, Scald, Headbutt, Zen Headbutt, Ice Punch, Iron Tail, Recover, Rock Slide, Rock Tomb, Ancient Power, Sludge Bomb, Sludge Wave, Water Pulse, Waterfall, Mud Bomb, Earth Power, Charge Beam, Fire Blast, Hex, Poison Jab, Thunder Wave, Yawn, Thunder, Curse, Gyro Ball.
Role identification: While Dunsire is nowhere near as overpowered as it was when I first made it, this version is still pretty darn good. Switching it into a Water move gives it a 100% chance to get Ancient Power boosts, which can turn it from "meh" to quite annoying very quickly. Recover gives it back any HP it might have lost before it used Ancient Power, and Hex works with the multitude of status-causing moves it has such as Ice Punch and Poison Jab. Curse + Gyro Ball is always nice, and Iron Tail has a 60% chance to lower Defense. Finally, Charge Beam has a 100% chance to give Dunsire a Sp. Atk. boost. All in all, it's still solid despite the nerfing.
Now for something completely different.
Parents: Claydol & Skarmory
Shared egg group: DNA
Offspring name: Claymore
New type: Ground/Steel
New base stats: 73/85/133/65/105/83
New ability and desc: Levitate + Sturdy = Floating Fortress: Just the effects of both abilities combined.
Notable moves: Rapid Spin, Defog, Stealth Rock, Spikes, Explosion, Light Screen, Reflect, Earthquake, Toxic, Roost, Iron Head, Iron Defense, Calm Mind, Roar/Whirlwind, Sandstorm, Swords Dance, Tailwind, Taunt, Torment, Charge Beam.
Role identification: The ultimate in entry hazard removal. It's hard to argue with an immunity to every hazard but Stealth Rock, but when you have a 4x resistance to the one hazard that can do anything to you, you have a nasty spinner/defogger. Access to hazards of your own isn't too bad either, and Roar supplements those. Dual screens make this thing nearly impossible to defeat, and provides support for the rest of your team. Tailwind, Taunt, and Torment are great support moves, and Sandstorm annoys any Pokemon that doesn't have an immunity to it. Calm Mind and Swords Dance give Claymore offensive presence, the former of which makes it even harder to take out. If you want a support Pokemon with many roles, Claymore fulfills your dreams.
On the opposite side of the spectrum, we have offense in spades.
Parents: Zweilous & Arbok
Shared egg group: Dragon
Offspring name: Arilous
New type: Dragon/Poison
New base stats: 76/95/80/75/85/79
New ability and desc: Hustle + Intimidate = Rampage: Effects of both combined.
Notable moves: Outrage, Coil, Gunk Shot, Crunch, Aqua Tail, Fire Fang, Thunder Fang, Ice Fang, Head Smash, Return, Superpower, Body Slam, Earthquake, Iron Tail, Infestation, Poison Fang, Pursuit, Seed Bomb, Sucker Punch.
Role identification: Zweilous has the strongest Outrage in the game thanks to STAB, Hustle, and a base 85 Attack stat. Now imagine that with a stronger Attack stat and Coil. You have just pictured a monster. This also most likely gives Arilous the strongest Gunk Shot, which is also extremely destructive. The Intimidate effect is very nice, giving Arilous an edge on physical opponents. Sucker Punch does a truckload of damage as well, and access to all the fangs takes out the majority of weaknesses Arilous has. Poison Fang's base 50 damage is boosted by the same combination that makes Outrage and Gunk Shot so scary, and a 50% chance to badly poison the opponent makes this forgotten fang more viable than it ever has been. If this thing gets off just one Coil, it becomes a sweeping machine that will leave a trail of broken dreams of victory in its wake.
Here's one that just seems all right.
Parents: Wormadam(Trash) & Volcarona
Shared egg group: Bug
Offspring name: Wormrona
New type: Bug/Steel
New base stats: 82/74/90/112/110/78
New ability and desc: Overcoat + Flame Body = Combustible: Grants immunity to weather damage and powder moves. Powder moves used on Wormrona deal 1/10th of the opponent's max HP as damage. This has a 20% chance of burning.
Notable moves: Flash Cannon, Iron Defense, Psychic, Shadow Ball, Quiver Dance, Bug Buzz, Stealth Rock, Struggle Bug, Amnesia, Calm Mind, Fiery Dance, Flamethrower, Giga Drain, Hurricane, Light Screen, Roost, Whirlwind, U-Turn.
Role identification: Combustible is helpful for dealing with Pokemon that use Spore or the like. Quiver Dance and Calm Mind serve to up the damage Wormrona can deal, and Amnesia and Iron Defense and to its longevity. Stealth Rock and Whirlwind work well together, and the rocks help deal with the Fire Types, which are the only Pokemon that pose a significant threat. Light Screen makes this thing even more annoying, and combined with Calm Mind/Amnesia, it becomes a Special Attacker's worst nightmare. Fiery Dance's high chance to up Sp. Atk is very nice, and Giga Drain/Roost provide reliable recovery. For such an unorthodox pairing, this isn't too bad.