It would be really cool if there was a champion that could decently ADC, support, mid, jg, and top. I don't know how it could be done, but there's always a way. There are champions who can play 4 of the 5 roles, but ADC is where they fail.
Edit: And they gotta be able to be bruiser or tank as well.
Their name is Kayle. She just does APC for the ADC role.
can carry from any (and every) role. Draven top holds the lane in glorious fashion. Draven mid is always the center of attention. Draven Jungle goes where his fans want him. Draven Support always makes sure Draven ADC goes full Draaaaaaaaaaaaaaaaven.
https://38.media.tumblr.com/ae402ec521e19176026a3334f63ddaf8/tumblr_n9692geiYG1r013jro1_400.gif
Jayce? Nidalee? Draven? Draven
Urgot ADC/Mid/Top are obvious. Support Urgot contributes poke, a slow, armor shred, and damage reduction. Not the worst support. Jungle Urgot is actually doable, but probably takes some Urgot expertise.
As others have said, this is Kayle's thing. She fits any role and builds whatever she wants and it's all good.
How do you even forget me? Do you need a reminder?
Although I can forgive you for forgetting me.
Isn't that Teemo ? At home at top, a strong mid laner. My friend and I always do support Teemo cause blind the ADC and win trades, adc teemo cause I did that and it worked amazingly and I even did jungle teemo, with my blind causing the monsters to miss and you can shroom your buff so that you always have them warded.
Wasn't that Riot's design philosophy with 
?
(except for ADC)
Nidalee
No one has mentioned: 
Hope you haven't forgotten the Crittlesticks.
could if she wasn't so bugged and unreliable. support could be a bit ify but i think it would work out.
A tank/bruiser who can also be a Marksman is going to be problematic: I don't think a Marksman should be allowed to have the inherent durability + crowd control that a tank requires.
The main issue with building something like this is that the only thing we have to rely on is itemization, and that has to be able to make the entire difference. I suppose you could fake a Marksman with a passive like "gain bonus Attack Speed based on your Attack Damage" and get their damage to marksman levels that way, but you'd have to similarly scale their CC with their total resists or their total health to avoid having a Marksman with too much crowd control.
I think it COULD be done, but the resulting champion would be very complicated, and, as such, I'd question if it's worth the effort.
can, but she's not very popular now...
That depends, though. By viable, do you mean strong in any position, or workable in any position?
a certain spooky ninja would like a word with you...
Her name was Irelia.
Riot feared her potential so much, they nerfed her even before release.
