I've played quite a few video games without a true formal currency. Even without such a currency, the natural behaviors of human beings in a barter economy would lead to the creation of a de facto currency.
In a barter economy, you judge how much an item is worth by how much you want it, how much other people want it, and how easily you could find a buyer for it. Assuming there exists some universally demanded item that moves quickly, that would quickly assume the role of currency.
If you look at trading card games for the iOS such as Rage of Bahamut, trade value is measured in small discrete units like healing items that restore stamina or battle power. Even though the economy may contain such disparate items such as cards, items, and other resources, something in nearly universal demand like the aforementioned healing items naturally becomes a standard unit of value against which all goods can be compared.
A video game economy generally avoids a lot of problems that real world economies have that facilitate barter, like the difficulty of enforcing contracts, trade goods that expire, or difficulties in transport. I daresay these factors would lead to the development of an in-game monetary economy along the lines I described above.