Unlocked after beating Level 3, the shop is where you can go to buy characters (like the Skater for example), upgrades, and costumes. Currently, the shop has 17 items:

Item Cost Notes
Respawn costs are reduced by 10. 1500 The first respawn becomes free!
Unlock the level editor. 500 This is not available on CoolmathGames, despite its recent update. This is not available on mobile either, but Player 03 plans to add it soon.
Unlock the Skater. 300 Also unlocked by beating Level 10.
Unlock the Lizard. 600 Also unlocked by beating Level 40.
Unlock the Child. 2000 Also unlocked by beating the Low-Power Tunnel.
Unlock the Student. 10000
Unlock the Angel. 12000
Unlock the Duplicator. 6000 Costs 25000 on mobile
Unlock the Pastafarian. 6000 Costs 25000 on mobile
Unlock the Bunny. 2000 Also unlocked by earning 8 achievements.
Unlock the Gentleman. 2000
Find more boxes in Infinite mode. 200*2n

(n= amount of times the upgrade has been bought)

There is no maximum amount that can be bought; it doubles in cost after each purchase. Also, curious how this helps you? Well, boxes cover holes, so it can reduce your chances of falling. Also, if you use the Duplicator, you can produce duplicates, or if you use the Pastafarian, you get more space on the light bridge.  Each one makes the boxes spawn slightly more frequently, but not by much. More importantly, the boxes continue to appear further in Infinite Mode, about 200m further per upgrade. The later upgrades to this are generally not worth it, unless of course every other item in the shop has already been purchased, and the player has amassed more power cells than could reasonably be spent on respawns making power cells almost worthless.

Boxes in Infinite mode will sometimes contain power cells. 2000 This means that if you get close to a box, you may get some power cells.
Unlock the winter costumes. 500 Also unlocked by completing the Winter Games.
Unlock the Halloween costumes. 750 Also unlocked by completing the Low-Power Tunnel.
Unlock the Pirate costume. 1000 Only appears in the shop when you have bought the Pastafarian.
Balance on razor-thin edges of tiles. 1000 Despite the most updated version of Run 3 on Kongregate and the Flash version of player03.com, this is only in the shop in the beta version of Player 03's website.
Unlock permission to refer to the Power Cells as batteries. 15 Unlocks the cutscene Batteries. Clicking on this in the shop when you have bought it shows the cutscene again.

This is what the shop page looks like when you first unlock the shop:

Shop Updated Images (Incomplete).jpg

And after you unlock everything (except for the "find more boxes" item):

Shop Updated Images (Complete).png

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