The relationships between the things on the character sheet.
A character sheet in a rpg is a list of Resources (things that are used and consumed: hit points, fatigue levels, sanity points, etc.), Effectiveness (skills, characteristics, abilities, advantages, etc.), and Positioning (who or what the character is in the fiction, where he is, in what situation and what does that mean, i.e. "King of Aquilonia", or "on higher ground than his opponent")
There things are related and exchanged all the time: being on a higher ground (Positioning) can give you a +1 on the "roll to hit" (Effectiveness) but trying to hit someone can use up your fatigue points (Resources). Or maybe you used up a fatigue point to roll on your "climb" ability (Effectiveness) to get on a higher ground (Positioning)
All there relationship and exchanges represent the currency of the game.
No matter if you are playing "freeform" or without character sheets, you still have a system, you still have characters, they still are positioned in a fiction, they still can do some things and can't do other things, and the things they can do (Effectiveness) can change their positioning, or the other way around. There is always a Currency.
Quick and Dirty
Characters take effective actions that (a) deplete, restore, build, or otherwise change their resources, and (b) change their circumstances, sometimes only immediately, sometimes profoundly. Characters' resources (a) provide breadth and depth to their ranges of effective actions, and (b) can also change their circumstances, immediately and/or profoundly. Characters' circumstances, of course, (a) constrain and provoke their effective actions, and (b) can change their resources.
A warrior kills his enemies, gaining experience and looting their wealth, but suffering exhaustion and wounds. For a while he's able to stay ahead of the curve - his experience does him more benefit than his wounds hamper him - but that time comes to an end. He uses his accumulated wealth to buy a wife and a home, and he turns his battle-cultivated experience to politics, to good effect. Now he's mayor.
Cycles of effectiveness, resource, and positioning, all trading off one into the next. That's currency. (So however it happens in play, that's your game's systemic currency, whether you have character sheets or dice or whatever or none.)
The exchange rate within and among Character Components. Currency may or may not be explicit (e.g. "character points"), but it is a universal feature of System, specifically as it relates to Character.
2010: Things on Character Sheets (2) (Vincent Baker)
2001: Character currency (attn: old thread, "Positioning" was still called "Metagame")