The bottom lane has two characters that are essentially joined at the hip for most of the game. While the support will eventually roam free and the carry may engage in some solo pushing or farming, those two characters spend much of the game together.
It makes sense then that these two characters should mesh well. Here's a list of my favorite teams. If you're stumped on whom to pick for carry or support, you should try some of these out.
Soraka and Nidalee: We'll get to the endgame!
Nidalee is one of my favorite characters. While she is a bit weird in the AD carry spot, she fills most of the requirements; she has good per-level AD scaling, amazing attack-speed scaling, an attack-speed steroid, and an unorthodox escape tool. Her steroid is also her heal, and while it's normally used on other characters, it's no big problem to cast it on herself purely for IAS. In addition, she has strong counter-push after level 6, so even a push composition won't do too well in the long term.
Adding Soraka's infuse to the mix lets Nidalee use her heal to sustain frequently as well as deploy traps and spears regularly in lane. Ganking a Soraka/Nidalee lane is almost impossible because both of them are constantly at full life and every bush anywhere near the two is going to be trapped. Additionally, once Nidalee has items, she cuts people apart. Nidalee scales extremely well to items because her base attack speed is absolutely ridiculous.
Soraka/Nidalee get to the midgame either ahead of or even with the enemy lane about 90% of the time. The only limitation is how good Nidalee's player is at last-hitting; Nidalee is hard to last-hit with early on, as her base AD is a bit low. A Soraka/Nidalee lane is also a bit weak until Nidalee starts getting items. A level 2 or 3, Blitzcrank/Vayne or similar comp can potentially deal a lot of damage before Soraka/Nidalee starts going, but if the two healers play it safe, they can get to the point that it is nigh-impossible to kill them.
Sona/Caitlyn used to be my favorite lane composition to play, although I started to realize its problems later on and don't play Sona much anymore. This lane combo is a lot more complex than Soraka/Nidalee, who simply play defensively and avoid confrontation whenever possible. The Sona/Caitlyn pairing is almost entirely reliant on both players predicting when the enemy carry gets complacent or the support makes a positioning error.
Caitlyn and Sona both harass extremely well. Caitlyn's massive autoattack range, coupled with Sona's health sustain and huge Power Chord poke damage, used to be the de facto standard in bottom lane prior to the arrival of Graves (and the Caitlyn nerfs).
Caitlyn's job is to last-hit as much as possible while actively denying the enemy the chance to participate in the lane. Her fantastic range lets her poke in relative safety, and Sona can assist in poking, mitigate counter-poke with her heal, and generally provide a lot of utility with her Power Chord. If Caitlyn gets in trouble, Perseverence and Celerity combined with 90 Caliber Net (shouldn't it be .90 Caliber Net?) can get her out of most problems. If Caitlyn is trying finish an enemy off, Crescendo and the Power Chord snare along with all of Caitlyn's abilities make kills much more likely.
More than anything else, this team is highly dependent on skill and is sort of bread-and-butter. It's not really a team you can hard counter (except with Urgot), but it's also not a team that beats anyone for free.
Soraka and Graves or Sivir: I can do this all day
This is probably my current favorite lane composition. I love playing Graves and Sivir, and when I'm one of those two, I really love mana. Graves can pretty much fight an enemy lane solo as long as he has access to his abilities, and giving him more of them by giving him Infuse is just scary. Every time I play Soraka/Graves, I always end up not spending enough mana and sitting at full because I'm normally kind of conservative about using Buckshot. In one of the more recent games I had with Soraka/Graves, I was up three kills by the 12-minute mark without needing to base because Soraka was taking good care of me (and the enemy carry was bad enough to let me kill him).
Soraka/Sivir isn't really the same thing, but it's the same basic principle: a character with ridiculously overpowered skills given nigh-unlimited mana by Soraka. Even if she's otherwise getting zoned out, it is very difficult to avoid getting harassed by her. Additionally, she can block some forms of aggression using her spell shield; I've done this against Blitzcrank lanes, and the results are almost always hilarious. In general, very aggressive lanes get ruined by Sivir if she's played reasonably well. Against a very passive lane, she generally can't make enough damage stick to force the enemy out.
I especially like recommending Soraka/Sivir because both champions are very cheap to unlock. They're also really strong against players you'll face at the lower levels of skill.
I don't normally run kill comps, but when I do, I run Leona/Vayne. Leona has ridiculous burst damage and a million ways to lock an enemy down. Vayne has ridiculous damage (burst or otherwise), crazy mobility, and one way to lock an enemy down. Once both ladies have their ultimates, it takes exactly one Zenith Blade hit before you're dead from full life. Every time I run this team, I feel bad for my opponents because my passive-aggressive style of play is very strong against the normal bullying that goes on in bottom lane.
Leona/Vayne is a kill comp, and just by running it you're telling your opponent that you want to slay his face. Every time I see a comp like this, I play conservatively at my tower. Actually, when I'm carrying, I generally just play conservatively at my tower because I know the enemy's going to be aggressive. Sadly, my Elo isn't to the point at which the enemy thinks being defensive is smart. I like Soraka/Graves better because the combo lets me act as if I'm struggling without actually getting behind, then suddenly turn on the killswitch and cutting loose. I can do that with Leona/Vayne, but it only works because my opponents are not taking my character's support seriously.
When I see an Alistar or Blitzcrank selected for support, I know the enemy team wants to kill people in lane. It's a telegraphed thing. I also know that if I play cautiously and poke defensively in response to last-hit attempts or positioning mistakes, I will come out ahead if my support is Taric, Soraka, or another healer.
Still, Leona/Vayne is my favorite kill comp entirely because of its massive blow-up-faces potential. Even a Blitz grab doesn't hurt as much as a Zenith Blade, even if the grab is safer. If Leona lands her snare, Vayne can move in and pick up the kill from almost anywhere in the lane. The reason this team works so well is that Vayne can be slightly out of position when Leona lands her snare and she'll still wreak havoc. If Vayne is in a good position to tumble into a wallslam, it's a done deal when Leona lands her E.
So there you are! Those are my favorite lane comps, more or less, and a bit of why they work. They're far from the only fantastic comps; there are quite a few, like Janna/Graves or Taric/Vayne, that are amazing and powerful. While you should definitely experiment and find teams that work well for you, it's important to keep in mind the strengths and weaknesses of each character. Some supports make some carries viable (Soraka/Nidalee needs Soraka to function), and others just synergize well (anything with stuns is better when you have a Leona or Taric). Good luck and have fun!
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