We introduce the Anonymous Post-Office Protocol (AnonPoP), a practical strongly-anonymous messaging system. Its design effectively combines known techniques such as (synchronous) mix-cascade and constant sending rate, with several new techniques including request-pool, bad-server isolation and per-epoch mailboxes. AnonPoP offers strong anonymity against strong, globally-eavesdropping adversaries, that may also control multiple servers, including all-but-one servers in a mix-cascade. Significantly, AnonPoP's anonymity holds even when clients may occasionally disconnect, which is essential for supporting mobile clients. AnonPoP is affordable, with monthly costs of 2 cents per client. It is also efficient with respect to latency, communication, and energy, making it suitable for mobile clients. We developed an API that allows other applications to use AnonPoP for adding strong anonymity. We evaluated AnonPoP in several experiments, including a 'double-blinded' usability study, a cloud-based deployment, and simulations.