Classical statistical methods have theoretical justification when the sample size is predetermined. In applications, however, it's often the case that sample sizes are data-dependent rather than predetermined. The aforementioned methods aren't reliable in this latter case, hence the recent interest in e-processes and methods that are anytime valid, i.e., reliable for any dynamic data-collection plan. But if the investigator has relevant-yet-incomplete prior information about the quantity of interest, then there's an opportunity for efficiency gain. This paper proposes a regularized e-process framework featuring a knowledge-based, imprecise-probabilistic regularization with improved efficiency. A generalized version of Ville's inequality is established, ensuring that inference based on the regularized e-process are anytime valid in a novel, knowledge-dependent sense. Regularized e-processes also facilitate possibility-theoretic uncertainty quantification with strong frequentist-like calibration properties and other Bayesian-like properties: satisfies the likelihood principle, avoids sure-loss, and offers formal decision-making with reliability guarantees.