You need to get 26 items in your inventory. This is normally impossible. Or is it?
It's hard enough to get 25 (a full inventory) so I'll explain that first.
This guide may spoil game contents, so please finish the storyline yourself before reading this.

Step 1:

Step 2:

Step 3:

Have 25 items in your inventory NOT including the tranquilizers, then pick up the Golden Dragon of invisibility. Exit the Gadgeteer's Shop at the Dock, then interact with the invisibility cloak. You will temporarily lose it from your inventory. You now have 24 items. Go to the Pharmacist in the Marketplace and rummage through his dumpster. You will obtain tranquilizers. You now have 25 items again. Go back to the Gadgeteer's Shop and enter. You will trip over the Golden Dragon of Invisibility and will be forced by the game to pick it up. You now have a secret 26th slot unlocked, with the item inside. Congrats!
This is how I got the acheivement, so your situation may vary.

• The crucial part is to have 25 items INCLUDING the invisibility cloak and to have some event you haven't activated yet to have an item to replace the invisibility cloak with.
  
  
• Then you just go back to the Gadgeteer's Shop and pick up the cloak to get your free 26th item slot.

Tips:

• Avoid finishing the gondolier event and irritating Spunky Goal. Otherwise you'll have to finish her event. You need to have access to Baby Goal and Lady Goal.
  Activate the "rebel hideout" and "platypus off the menu" events by talking to Lady Goal and Baby Goal about how to convince them to have surgery done.
  
  
• Don't combine items unless you end up with 25 and need to make room for the first time you pick up the Golden Dragon of Invisibility.
  
  
• Don't combine the hook and thread. The only thing you need to give the Fisherman toward this acheivement is the fortune cookie. That way you can start hunting Platypus eggs.
  
  
• Collect only the Earth and Water Platypus Eggs. Collecting the other ones requires combining items or unloading your inventory.
  
  
• Unscrew and take apart the shredded sunglasses of eternal darkness, after you juice the Cucumber of Revelation (requires activating rebel event and talking to Garloff. Take his bathroom key and go into the bathroom. Give the toilet paper to the pharmacist) You will now have 3 items in your inventory instead of 1 (you gain 2).
  
  
• Make sure you go see Captain Seagull in Bellevue and get the memos to give to Crane. This will give you about 5 slots filled towards your acheivement. Make sure you grab the urn, too because, well, YOU CAN!! Muahahaha

Just follow my example provided. You don't need to be going out to get the tranquilizers, it could be something else, like the hook from the fisherman at the dock.

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