With 7 extra spaces, you have a lot of options to fill up the board. Here are some archetypes you can look toward building. We'll also show some strong skills you may encounter.
In general, this build gets a lot of its strength through skills and not pure upgrades. Keep an eye out for pertinent Skill encounters to upgrade your board.
Burn
As expected, you can always use more slots for other Burn items. There are other utility items that Burn while also doing other things, but having more Burn won't be bad. This will be great if you have Skills that buff the Burn on all your items, giving every piece more value.
Final Flame (Gold) is a starting skill that you can pick to snowball the early game. The hard part is trying to get both a Burn item and Final Flame on Day 1.
Slow
Lighthouse: Burn whenever you Slow an item.
Slow is another utility that synergizes well with Burn due to the Lighthouse. This is a property that burns the enemy every time you slow them. Slowing enemies down is also important because your Burn needs time to ramp up and defeat them.
With the Lighthouse, you can look to use a variety of other Slow items such as Clamera, Tripwire, or Dock Lines.
Haste
The Burn archetype also has great access to Haste through the Trebuchet. The Trebuchet will Haste an item whenever you Burn the enemy. Since Weather Glass has Multicast, every cast will Haste an item. After a bit of ramp up time, your entire board can be Hasted.
There are other ways to obtain Haste through other items or skills as well. Trebuchet is very common, but you can look for things like Astrolabe as well.
Ammo
Pop Snappers: Burn on a 3s CD
Incendiary Rounds: Adjacent Items Burn on use & have 1 extra Max Ammo.
Lastly, Ammo items are also solid items to incorporate into the build. Rainbow Potion is an Ammo item, so getting additional Ammo through Incendiary Rounds or Cannonball are great for using the Potion over and over.
Pop Snappers is also a great early item to get off quick Burns. If you're using Trebuchet having a source of quick Burn to get the Haste rolling is important for tackling faster Builds.
This skill is a random drop from Master Alchemist, giving you another reason to pick him to fight. With this, you can use powerful Potions like Rainbow Potion as long as you place them next to Weather Glass or a Lighter.
Port / Potions
Another variant for the build is to go all in on the Port. With the Port, you essentially get unlimited Ammo on your items, and the Port itself will randomly generate very powerful potions for you to put in.
This can be a strong path if you naturally upgrade your Port quickly and start getting powerful Potions. Since the Port is Large, we can't easily fit in Haste through things like Trebuchet. Try to find another source of Haste either with Enchantments or Skills.
There is a 5 second Potion for every ailment (Burn, Poison, Freeze, Slow). Using these in combination is very powerful and the Port can charge these Potions faster too. You don't need the Port if you have other sources of Ammo or Reload.
If you have lots of Ammo items on board, Heated Shells is a great source of additional Burn.