Online learning is not always about memorizing everything. Since the future can be statistically very different from the past, a critical challenge is to gracefully forget the history while new data comes in. To formalize this intuition, we revisit the classical notion of discounted regret using recently developed techniques in adaptive online learning. Our main result is a new algorithm that adapts to the complexity of both the loss sequence and the comparator, improving the widespread non-adaptive algorithm -- gradient descent with a constant learning rate. In particular, our theoretical guarantee does not require any structural assumption beyond convexity, and the algorithm is provably robust to suboptimal hyperparameter tuning. We further demonstrate such benefits through online conformal prediction, a downstream online learning task with set-membership decisions.