There are several champs that you can't pick just because they're banned so often, like nocturne, zilean, shen, and eve. My five would be:

Warwick- most versatile jungler, never banned.

Kennen- rarely played champion that can be viable as ap, ad or tank, depending on how you build him

Sion- again, really flexible, can fill multiple roles (even jungle in a pinch) depending on how you build him

Alistar- another flexible champ (getting my drift?) who can play either full tank, dps/tank hybrid, or ap. Aoe disables ftw.

Ashe- her ult is just too good to not be able to use

Nidalee - support
Rumble - AP soft
GP - AD hard
Nasus - Tank build
Kass - pwnage

Xin Jungler/Solo top
Nocturne Jungler/solo mid
Maokai Duo bot
Vayne Solo mid
Zilean Duo bot

Soraka - Best healer in game
Udyr - Have you seen my udyr videos?
Nasus - Great Off-Tank and late game carry
Corki - Just cause of this guide http://www.youtube.com/watch?v=mWbwypd1pM4
Xin Zhao - For right now, he is really good this patch

Sion (Bread and butter. He is just so good)
Ashe. (Great carry. Good for backdooring and killing towers late game)
Sona (A good healer will keep the team punching all night long)
Mundo (In my opinion the best tank)
Karthus (Who do not want to kill opponents from the other side of the map)

Jax - he's a great laner and he does massive damage midgame onwards.
Kennen - my 1st champ and very flexible
Shen - I cant count how many times I've taunted 3 or more champs and had my team wipe them out.
Vayne - a ranged jax imo, even better at laning than jax
Taric - I like to play support Taric but I like being able to tank and heal.

Udyr: Melee/Jungler/Offtank/TankyDPS. Many builds. High skill cap. Monstrous late game. Survivable.
Irelia: Melee/Assasin/TankyDPS. Many builds. Good all around. Unpredictable. Survivable.
Ezreal: Ranged/Carry. Multiple builds. High skill cap. Strong carry. Survivable.
Alistar: Melee/Tank/Roamer/Support. Multiple builds. Very strong and good on any team. Survivable.
Soraka: Ranged/Support. Multiple builds. High skill cap. Tied with Janna for strongest pure support. Tryhard not to die.

Singed - Essentially requires you to play insane. Probably one of, if not the, funnest champs I've ever played.
Ashe - Dat utility, dat ult, dat carry
Janna - Incredibly fun to play, as well as being one of the best supports ever. Nothing beats a free BF sword every ten seconds.
LeBlanc - I haven't played her in a while, but she is fun and just... so bursty.
Jarvan IV - My favourite tanky dps, he is incredibly strong while being totally unkillable.

Amumu (jungle tank, absolutely love this guy, my best champ by far)
Ryze (solid mage)
Xin (melee dps/possible jungle)
Zilean (my favorite support)
Ashe (straight up carry)