We consider discrete-time uncertain processes with finite state space and study the properties of game-theoretic upper expectations developed by Shafer and Vovk. We start by proving some basic properties, e.g. monotonicity, law of iterated upper expectations,... Afterwards, we show that this operator satisfies several, more involved continuity properties, including continuity with respect to non-decreasing sequences and continuity with respect to specific sequences of so-called `finitary measurable' functions, which are functions that depend on a finite number of states.