How to cause a chrome app to update as soon as possible?

Install a listener for chrome.runtime.onUpdateAvailable, which fires when the new .crx file has been downloaded and the new version is ready to be installed. Then, call chrome.runtime.requestUpdateCheck:

chrome.runtime.onUpdateAvailable.addListener(function(details) {
  console.log("updating to version " + details.version);
  chrome.runtime.reload();
});

chrome.runtime.requestUpdateCheck(function(status) {
  if (status == "update_available") {
    console.log("update pending...");
  } else if (status == "no_update") {
    console.log("no update found");
  } else if (status == "throttled") {
    console.log("Oops, I'm asking too frequently - I need to back off.");
  }
});