A system can include at least one check valve which prevents outward flow through a wall of a tubular string in a well, the wall surrounding a longitudinally extending flow passage of the tubular string. The check valve can include a ball or poppet which sealingly engages a seat. Another system can include multiple redundant check valves which prevent outward flow through a wall of a tubular string in a well. A well tool can include multiple redundant check valves which prevent flow in one direction through the wall, and which permit flow in an opposite direction through the wall. Each of the multiple redundant check valves can include a ball or poppet which sealingly engages a seat.