First, because "enjoyment" is a complicated measure, unlike, say, temperature, we first have to model it. What does "enjoyment" mean? Once we've figured out what we mean by "enjoyment", then we can measure it.
One model is the Affect-Behavior-Cognition model (hey, ABC) proposed by Nabi and Krcmar (2004).
Another model is GameFlow (Sweetser and Wyeth, 2005 (pdf download)), which has 8 parts: concentration, challenge, skills, control, clear goals feedback, immersion, and social interaction. These map relatively well to the 8 components of Flow. (memo to my future self: Pervasive game flow.)
Vorderer, Hartmann, and Klimmt (2003) note that competition is an important part of the enjoyment of video games, although they also admit that there may be other components (challenge, fantasy, curiosity, and others).
Fang et al (2008) put together an 11-part questionnaire to measure enjoyment along the Affect, Behavior, and Cognition axes.
Ravaja et al (2006) catalogued some measures for valence and arousal of emotions: heart rate and GSR for arousal, EMG cheek and around-the-eye muscles for positive emotions, EMG brow muscles for negative emotions. (see the intro; the paper itself is about opponents, which I'm not so interested in)
Mandryk and Inkpen (2004) consider even more physiological measures in their study of opponents: EMG jaw, heart rate variability, and respiratory measures. Again, their particular study is not so interesting to me.Hazlett (2006) did a more detailed study on the cheek (zygomaticus major) and brow (corrugator supercilii) and found that they did indeed reflect positive and negative events, respectively, in games.
Andersen et al (2011) simply tracked how long people played a game under various conditions and how often they returned.
Seems like the questionnaire measures exactly what people want in games (enjoyment) but through the often-cloudy lens of self-report. The physiological measures seem good for emotional state, while the time-on-game measure is more suited for engagement. All good measures; I'm sure there are many others too.