I'm not completely a free-to-play player, but I certainly haven't spent much money on this game ($20 on packs, $10 on arena runs). As a result, I fairly often find myself in a situation like this while I claw my way up the ladder:
Turn 8: Opponent plays Ragnaros. I have no counter.
Turn 9: Opponent plays Faceless Manipulator to copy his Ragnaros. I destroy my underwear.
So, I've been working on a good deck using the cards I have from my not-even-nearly complete collection. I've come up with Buttwing, and I think he really has something going for him (even if it's not a Legendary).
He has two methods of winning:
1. Trade minions, board clear and dispel constantly through the early game. Then, fill up your board again with 2-for-1s and totems, and bloodlust the opponent for massive damage.
2. Rely on periodic semi-beefy attacks--6 damage here, 10 damage there, then one turn windfury + rockbiter weapon into 18 damage from one guy.
The first, so far, has worked well against aggro players who are forced to trade against Buttwing's onslaught. The second, so far, works well against passive players who draw the game out into some huge finale (turtle Mage, Jaraxxus Warlock, any Druid...)
If you guys want some fun without a deck full of legendaries, give Buttwing a try. Also, post tips if you think there's a way to make him better.
Here's the decklist: http://www.hearthpwn.com/decks/28825-buttwing-the-lusty-shaman
And here's a video of Buttwing in action: