How to remove event listener in Chrome extension

removeListener takes an argument. You need to name the listener function and then remove it by name:

function doStuff(request){
    chrome.extension.onRequest.removeListener(doStuff);
    other_function(request);
}
chrome.extension.onRequest.addListener(doStuff);

Or, more succinctly:

chrome.extension.onRequest.addListener(
    function doStuff(request){
        chrome.extension.onRequest.removeListener(doStuff);
        other_function(request);
    }
);