This is a Jungle Giants deck with multiple win conditions:

1. versus aggro: if you manage to not die the turn after you complete your quest, most opponents give up as you start putting down multiple big taunts per turn
2. versus mid-range and light control: build a big board, go face; surprise damage from magnetic minions and Alexstrasza helps
3. versus heavy control: Mecha'thun

The most commonly used win condition in my experience is #2. Mecha'thun is used often enough to be worth having in the deck but it's not the primary win condition.

Although there are several legendaries in the deck list, the only one that is essential is the quest. See the last section for replacement suggestions.

Mulligan and Early Game

Oaken Summons is the best card in the early game, so always keep that and toss most other cards when looking for it. Cursed Disciple gives you two ticks on your quest, so that's also a keep. The other 3 and 4 cost minions are probably keeps, unless you foresee that they're not good against a particular opponent.

I usually don't keep Wrath and Ferocious Howl, since I consider them more of a consolation prize: they're fine if you draw them, but not the big swings you're looking for. You could keep Wrath if you expect aggro or already found your 4-drop(s).

Archmage Vargoth is great when pulled by Oaken Summons, so it can be worth not keeping him if you have or are betting on finding Oaken Summons. If Vargoth is in the deck and you are on the coin, it's best not to coin Oaken Summons on turn 3, since you'll have a 50% chance of Vargoth whiffing.

Completing the Quest

Roughly turn 4 to 8 is playing minions on curve, if you can. If not, draw cards. Ideally you want to have some card draw left after you complete the quest to get more of those 0-cost minions, so don't use all your draw unless you really have to.

Grizzled Guardian is usually two ticks on your quest, since the only minion it can pull before completion that doesn't have 5 attack is Vargoth.

Zilliax is worth delaying unless you're in danger of dying, since later in the game you can magnetize him on a big mech and gain a lot more life.

If you have Barnabus the Stomper in hand, be careful with Oondasta, since Barnabus is a beast and the battlecry doesn't go off if he's pulled.

There are two copies of Naturalize in the deck, so you can freely use one to keep the opponent's board in check. There is only one copy of Innervate though. Don't hesitate to use your last copy of either of them if you need them to stay alive: Mecha'thun isn't going to save you if you're dead. Games against decks that pressure you early are probably not going to last long enough for Mecha'thun to be relevant anyway.

Post-completion

Now it's time to start keeping track of Mecha'thun. On any turn, the possible situations are:

• Mecha'thun is in hand and costs 10
• Mecha'thun is in hand and costs 0
• Mecha'thun is in the deck and costs 0
• Mecha'thun got burned or pulled or otherwise removed

If you're sure you're not going no need Mecha'thun to end the game, you can just play him as a minion (in particular if he costs 0) or pretend he doesn't exist. Be careful though that for example a game vs Hunter can turn into a control match just by your opponent playing Deathstalker Rexxar.

If Mecha'thun is in hand costing 10, you have to keep Innervate. Otherwise, you can play Innervate if that helps you fit multiple expensive cards into a turn.

If Mecha'thun is in the deck, he costs 0, which means he can get pulled by Oaken Summons and Grizzled Guardian. Whether that is good or bad depends on how you predict the match to go.

Despite having to keep Mecha'thun in mind, your primary goal is to win in this phase of the game, not just stay alive. Mecha'thun is plan B.

After a few turns you'll probably be able to flood the board with big minions, but how many should you play? If you have Mecha'thun left as a win condition, you don't have to be super careful to not over-extend, but still it's better not to throw good minions away. In general, it's good to threaten lethal, since that can force your opponent to make sub-optimal plays.

You can get surprise lethal regularly by having for example 10 power on the board and playing Alexstrasza plus Wargear or Ultimate Infestation. Naturalize or a rush minion can deal with taunts.

Grizzled Guardian doesn't look threatening, but opponents often underestimate that it pulls 10 attack from the deck. Especially Damaged Stegotron is huge when its battlecry doesn't go off.

I typically keep 3 big minions on board, or one plus Grizzled Guardian. If your opponent deals with one of them, you still have enough power left. And if your opponent uses AoE, you can build several equally strong boards in a row.

Try to play any non-discounted minions when you can: these are less flexible and they can become a bottleneck going into the end game.

End game: Mecha'thun

If you're nearing the end of your deck and no victory is in sight, you should start planning for Mecha'thun.

If you haven't played Ultimate Infestation yet, you should do so soon, since it is very painful to play this with an empty deck. If you have all the essentials in hand (Mecha'thun, Naturalize and, if Mecha'thun costs 10, Innervate), it's fine to burn a few cards.

Make an estimate of how many turns you need to empty your hand and how many turns before you draw the remainder of your deck. If your deck is going to run out first, then use Branching Paths for armor and play Oaken Summons and Grizzled Guardian as late as possible, so they pull fewer cards from your deck.

Besides having an empty deck and hand, Mecha'thun also requires an empty board. Usually this is not a problem, because your minions have a lot of attack, so ignoring them is dangerous. Also many have taunt, so your opponent cannot easily go past them. However, if your opponent also has a big taunt wall, it can take a while for all your minions to die. So you might have to make some inefficient trades, but be careful not to do that too soon or your opponent might realize that Mecha'thun is coming.

If you have two copies of Naturalize left, you can use one to clear your own minion, except if it's Evasive Feywing or Winged Guardian. Winged Guardian is especially resistant because of its high health, reborn and being untargetable by spells, so try to play these early in the game.

Replacements

While the core of this deck has been unchanged for a long time, I keep adding and removing single cards to see if it can be improved.

Turn 2 and 3 are weak points of the deck. You could put in something like Tar Creeper to slow down aggro, but at the same time you're slowing down your quest completion. Putting in spells can work around the minion restrictions, but since spells are not getting discounted by the quest reward, you are then weakening the mid-game and slowing down Mecha'thun.

Ramp seems like a natural pick for a deck that is slow to start, but the problem is that most of the ramp is too slow. For example Wild Growth was good at 2 mana, but at 3 it doesn't really help that much, since we have many good plays on 4 mana. Greedy Sprite is even worse, since opponents often don't attack into it, so its reward comes on turn 5. Breath of Dreams might be an option, but the deck would have to be changed significantly to put in enough dragons to make it viable.

At first I was concerned that King Mukla would give the opponent too much of an advantage, but since the bananas cost 1 mana for +1/+1, they're neutral tempo-wise. And tempo is all we have to be afraid of in the early game. Similarly, don't shy away from using Naturalize on a high-attack minion.

I had Elder Longneck in an earlier version of the deck. It's an OK card when played from the hand, but pretty bad when pulled from the deck, so I dropped it. I don't think any of the other 3-mana minions are good enough to make the cut.

For 4-mana minions, I've tried a lot, but the current selection worked best for me. I'm undecided about Steel Rager vs Evasive Feywing: the latter is better against multiple small minions, but the former is more proactive and more flexible in the late game. So for now, I'm running one of each.

Taunts are essential to survive until the quest reward kicks in, so if you want to make replacements, I suggest keeping the number of taunts the same or even raising it.

Alexstrasza helps a lot in getting wins, so I think she's the second most important legendary in the deck. You could replace her if you lack the dust to craft her, but I don't see any other reason to leave her out.

If you don't have Mecha'thun, you could put in other late-game threats like Chef Nomi or Hadronox. Maybe substitute some big deathrattles and run N'Zoth, the Corruptor. Or perhaps just add a copy of Savage Roar to get more surprise lethals.

Zilliax is just great against aggro. Later in the game, magnetizing it can heal you a whole lot, but sometimes the 3 extra damage is also relevant.

Oondasta is useful to get non-discounted beasts out of your hand for free, but it's not a must-have by any means.

If you don't have Ultimate Infestation, you could replace it with Overflow. If you don't like the risk of drawing it late and either blocking Mecha'thun or taking lots of fatigue damage from playing it, Nourish is a safer alternative, but a lot less efficient. The 5 damage from UI has delivered lethal for me many times, while the 5 armor is always useful.

I don't have Kun the Forgotten King, but that would probably be a good card to add if you have him.

Here are some big taunts that you can fill holes with. These were in earlier versions of the deck and performed decently, but were out shined by newer cards:

• Dark Arakkoa: one more health than Stegotron, but without the pull and magnetize upsides
• Furious Ettin: just good stats

Some cards that I tried but didn't really work for me:

• Emperor Thaurissan: only has pseudo-taunt, so dangerous to play on-curve versus aggro; in general Innervate accomplishes almost the same thing and is more flexible
• The Lich King: solid taunt, but the cards he gives you are not all good for this deck, especially if you're working towards Mecha'thun
• The Curator: not great when played before quest completion: doesn't advance the quest and draws you non-discounted cards
• Amani War Bear: rush is nice, but on a taunt it makes less of a difference than usual, since the opponent is often forced to trade anyway
• Malfurion the Pestilent: good defense, but doesn't get discounted, so I'd rather have a big taunt
• Swipe and Spreading Plague: good versus aggro and can be used offensively post-completion (buff scarabs with Branching Paths), but as a spell it doesn't get discounted, so it doesn't synergize all that well

Maybe I'm too greedy by leaving out the death knight and spells, but this is a deck I play for fun and while I like winning, it's not a deck I would pick if winning was the only thing I cared about.