I feel like I've never come across a Champion that can play all lanes (Top/Jung/Mid/Bot) successfully, sure you could say 
or 
or even
could play all lanes. However this would take tremendous amounts of "Player skill" so to speak. Does anyone have any champs they think can suite the category of a "Quadruple Threat"?
Kayle can. Easy.
She has a strong kit that is good in all lanes, even jungling.
Hell, you can even ADC or Support with her. Also, the people who are good with her are terrifying.
Fiddlesticks.
Kayle is the obvious pick, but have you considered...
little Satan?
Kennen top. Kennen mid. Ranged champs can jungle abusing leash mechanics (wouldn't recommend it)... Kennen ADC is actually pretty decent. Kennen support actually got pro play last year (decent manaless poke and engage)
: All 5 roles
: top, mid, support, ADC(not optimal, but doable)
: top, mid, support, and jungle
: top, mid, jungle, support(I personally think the fact that he requires CS to refill his puny mana pool makes him a terrible support, though)
: could go top or mid, but only recommended if you're team comp accounts for it.
: top, mid, support, ADC
: top, mid, ADC
: jungle and top, viable mid and support as a counter pick to to melee mids/supports.
As a matter of fact, most champions are viable in any lane besides the jungle "lane" if the counter is hard enough or you can effectively abuse the opponent not expecting or knowing how to fight the matchup. My first game as Singed, the enemy team switched their support Soraka with their original top laner.
The fun thing about Tristana mid is that your opponent wont know whether you're going AP or AD until the match starts, meaning they'll have to guess what defensive runes/masteries to use.
I think there are more than this, but Kayle is amazing anywhere. Yasuo and Ez(and GP sorta) can do pretty much whatever they want(though I'm not a big fan of Yas top).
Lee Sin
teemo can do it all maybe twisted fate too idk how well his support would work
Top, Mid, Jungle, ADC. Twitch and quinn are known junglers, but Vayne can surprisingly jungle with her crazy kiting. Twitch top actually saw competitive play I believe.
All 5 roles. I've seen Lulu jungle work.
no one brings up j4? he can top.. jungle and support idk bout mid he is best as a jungler but if your jungler decides not to pick him, you might as well snag him for some other role j4 is just a very strong champ right now, arguably more useful than the likes of kayle and fiddlesticks and kennen or watever thats been suggested
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I feel like I've never come across a Champion that can play all lanes (Top/Jung/Mid/Bot) successfully, sure you could say
All lanes? Yes. All positions, not really.
, etc
he can top, mid, jungle, and support
twisted fate- top lane, mid lane, jg , adc and support
support cause he got slow and stun
work in all 5 roles.
morgana jg worked in s4 really well, in s5 iunno
marksman, jungler, mage, "support", he can be built many ways and do many things
Urgot
Slows, stuns, sustain, shields, mid laner, top laner, support, hell even jungle, tank, bruiser, etc. Tahm Kench has it all.
Think more "role" than specific lanes, and that's the key to breaking the mold/meta here.
Cho'Gath is a prime example of a melee with ranged abilities who can build anywhere between tanky and AP. With his sustain, he can easily jungle, go against tanks/bruisers, assassins and mages, and sometimes I can see him supporting and doing just fine.
The roles are more important than the lanes. Picking a Riven top with a Shaco jungle isn't always the best idea, for instance, even if that's where they are "meta."
Mid lane, from my experience, tends to be the snowball lane and holds champions who scale the best with levels (assassins to hit 6 and other power spikes to, well, assassinate. They don't do as well in other roles because their power spikes are delayed. Mages for base damages as well). Utility mages, however, are popular in mid as well as support, and a few go solo top since they aren't as reliant on levels and they have enough utility and damage to solo lane as well as teamfight. Lulu used to be like this and can be still with team comps such as Juggermaw, and other champions.
Are you forgetting about the god, the king, the savior 
"MUNDO GOES WHERE HE PLEASES!"
and are my picks. They can both play all lanes and jungle. Out of all of the roles between the two mageplank is the iffy one. If you can get the money though...
noone has said pantheon yet, he is a very unconventional support pick, and i mean technically you could build him glass cannon carry if you are so far ahead that there isnt much of a threat on the enemy team
Jayce, can effectively top, mid, jg, and adc. I'm not sure about support, but he is pretty OP in any lane, although he does require a bit of skill to master.Thresh can be played any role. Support? Obvious. ADC? Heck yeah. Mid? Really strong if you know how to do it. Jungle? It works pretty well. But his best lane by far? Top. Farm for days and 2 shot anyone and then 1v4 the rest of the enemy team until your team arrives. He's the opposite of Nasus, with Thresh you farm for Tankiness from souls, (and damage from them) and build pure damage and MR.
- really good any where, and can fill almost any role.
- Can technically head into any lane, but in bot lane he should be the support. Yes, jungle donger is a thing.
- balanced on a razor's edge, you will either dominate your lane or get dominated. Either way, you decide the outcome of the entire game. Not for the faint of heart, requires mastering.
His toxic can pretty much do for anything, in any lane. Q-Q LOL
Pretty much every single role is viable. Also definitely.
Fid can top, mid, support, jungle, and even adc if you are feeling crittlesticks lol.
can, it just takes a bit of skill to jungle successfully. She used to be a much better adc, but can still survive...
Jarvan IV
Yasuo is possibly the best at every lane.
Top, Mid, Support; with shield and passive could be a weird off-meta jungler maybe (but even I'm not crazy enough to try that)
