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Level 2: Looking for Larvae in All the Wrong Places
Adventure in the Spooky Forest until you come across a mosquito larvae (probably very soon), and bring it back to the Council of Loathing. If you do not have one already, they will give you 500 meat to buy a Familiar-Gro Terrarium to put the larvae in. Once in there, it will hatch into a mosquito for you to use.
Level 3: Ooh, I Think I Smell A Rat
In the Typical Tavern, in the Distant Woods, adventure in the basement darkness until you encounter the rat faucet. Turn it off to end the quest, or continue adventuring to find Baron Von Ratsworth and his special items.
Special Side Note:
It might be prudent to get the Spooky Temple for faster levelling up, especially if you are trying to ascend as fast as possible. Adventure in the Spooky Forest until you get the Monolith adventure. Click "Copy the Map". Now adventure to get the Spooky Sapling and Spooky-Gro Fertilizer. Use the Spooky Temple Map to gain access to the Spooky Temple in the Distant Woods.
The additional Level 4 Quest:
Slay Your Nemesis
Before heading to the Dark and Dank and Sinister Cave, collect:
Gnollish Flyswatter, Cog, Sprocket, Fairy Gravy, Knob Goblin Tongs, and Catsup (from the hermit)
Equip the flyswatter to get past the fly bend, walk past the door (to use the cog and sprocket), give the troll your fairy gravy (just talk to him), equip the tongs to get past the salad-covered door, and give the troll the catsup (or ketchup).
Now you can fight your nemesis and collect the special item.
Level 4: Ooh, I Think I Smell A Bat
Before beginning this quest, you need stench protection. Either combine two bumcheeks from the Sleazy Back Alley to make an asshat, or adventure in the Entranceway of the Bat Hole until you get a Pine Fresh Air Freshener, and equip it. Now adventure in Guano Junction until you get 3 sonar-in-a-biscuits (or just buy them in the Mall). Use all 3 to knock down the walls in the Bat Hole. Now fight the Boss Bat and get his bandanna!
Level 5: The Goblin Who Wouldn't Be King
Adventure inside the Cobb's Knob Harem until you get the harem veil, harem pants, and Knob Goblin perfume. Equip the two items and use the perfume to complete the Harem Girl disguise. Kill the King!