I would say
Lulu,
Kennen, and maybe
Urgot are the only champions who can be played in all three lanes competitively, and I feel like
Kennen is the strongest of the three.
Sion and
Pantheon also deserve mentions for being decent at top, mid, and jungle. Kayle and Gangplank are both extremely versatile champions that could have filled any role in the past, but their mana costs are very restricting and make them suboptimal in most lanes now (Gangplank Q, Kayle Heal).







Ad: he can play AD perfectly, but u should know that.
Mid: he can near 1-shot any mid.
Top: he is one of the best top champions as he has hard cc, sustain and a great escape ability.
Support: as support he can harras them to death and has enough cc to be viable, his mobility will help on warding.
Jungle: since the new jungle everybody can jungle, seeing he has great aoe for clearing, mobility for initiate and cc it could really work. Never seen or tried it actually, but I'm going to soon now :).

AD Carry (solomid, duo bot lane): Decent range although CDR dependent, Great damage, ulti + BT let's her go back to full health, has a chaseability and runnability, usually a lategamer (like most carries)
Hybrid Carry (solotop): Great overall dps, mid to lategamer
Support: It works and I've seen it work quite often. How can she support? Ulti makes carry invincible for 2-3 seconds, W heals and helps carry chase, run, Q helps carry chase, run, E shreds enemies defenses. Support Kayle usually builds onto a tank build. She makes for a kill lane, so bad support Kayles will often turn it into a stale/passive lane where they'll run out of W fast. Best carries for her are mobile ones, like Vayne, Caitlyn and Graves, with a mention of Ashe because of permaslow.
Tank: The worse Kayle build IMO, because she barely has any ability to initiate. However, can dish out decent shreds while healing the enemy team's target or ulti'ing said target.
AP Kayle: Not a fan, specially coz it requires attack speed items either way. Would rather full hybrid;
On-Hit Kayle: Another great build, synergizes with passive and is very similar to her hybrid build.
Jungler Kayle: It works, whatever people say. Clears jungle fast, can frequently survive invading junglers or kill them if they're careless (udyr is the only trully annoying enemy jungler that can destroy her due to his chaseability), ganks with huge damage + her slow. Just leave the bush with W and Q them. Can dive or come from opposing tower once she reaches level 6. Requires people to have eyes on their faces though, or some sort of communication tool, like skype. Also a great lategame carry.
(minor edit) There are also other annoying counterjunglers (just like you can be one, although her splash doesn't usually allow to leave that troll leftover wraith if you're into that). But I love her anyway <3
Dat guy above nevef tell lies.
Im fact, Kayle jungles almost as fast as Shyvana and Malphite, and is just as fast as Nocturne. Her ganks are INSANE as she can Towerdive liek an extremely badass person.
And Kayle scales into late game really good after the rework. "We Nerfed her base stats and boosted her scalings to make her less of a threat late-game." Lol Riot.
Im fact, Kayle jungles almost as fast as Shyvana and Malphite, and is just as fast as Nocturne. Her ganks are INSANE as she can Towerdive liek an extremely badass person.
And Kayle scales into late game really good after the rework. "We Nerfed her base stats and boosted her scalings to make her less of a threat late-game." Lol Riot.

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Jarvan IV yet? Man, are you guys unmanly ;)
J4 is great. Top, jungle, bot support...no problem;
Tank, tanky dps, bruiser, dps support...no problem;
shield, slow, knock up, as buff, and slam dunk of death? Hell yeah.
And every AOE team will love you.
J4 is one of the examples of a versatile jack of all trades. In no role he is considered to be the sh*t, but he can adapt to nearly any team and can change his role quite quickly if needed.

J4 is great. Top, jungle, bot support...no problem;
Tank, tanky dps, bruiser, dps support...no problem;
shield, slow, knock up, as buff, and slam dunk of death? Hell yeah.
And every AOE team will love you.
J4 is one of the examples of a versatile jack of all trades. In no role he is considered to be the sh*t, but he can adapt to nearly any team and can change his role quite quickly if needed.
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Nobody thinks Lulu Janna and Nautilus deserve to be mentioned? Any champion that can perform various roles and various lanes belong in this thread.

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Other that come to my mind would be Lee Sin, Janna and GP.
Taric, maybe?