We present the analytical and empirical probabilities of a player achieving a single-player high score after playing a series of games. Using analytical probabilities, simulated game data, and actual game analytics data from two popular mobile games, we show that the probability of reaching a high score decrease rapidly the more one plays, even when players are learning and improving. We analyze the probability of beating the previous k scores, placing on a Top m Leaderboard, completing a streak of k consecutively increasing scores, beating the mean score, and introduce a metric called “decaying high score” that is parameterized and easier for players to achieve. We show that players exhibit different types of learning behavior, which can be modeled with linear or power-law functions–but that in many conditions skill improvement happens too slowly to affect the probability of beating one’s high score.