Firefox: don't remove finished downloads (browser.download.useToolkitUI = false), code for Custom Buttons extension https://addons.mozilla.org/addon/custom-buttons/ ("initialization" section)

dontRemoveFinishedDownloads.js

// https://gist.github.com/Infocatcher/6452231
// More: https://github.com/Infocatcher/Download_Panel_Tweaker
// (c) Infocatcher 2013

var retentionDays = 21;
function dontRemoveFinishedDownloads(patch) {
	// See https://github.com/Infocatcher/Download_Panel_Tweaker/issues/5 for details
	try { // Firefox 26+
		// http://mxr.mozilla.org/mozilla-central/source/toolkit/components/jsdownloads/src/DownloadIntegration.jsm
		var {DownloadIntegration} = Components.utils.import("resource://gre/modules/DownloadIntegration.jsm", {});
	}
	catch(e) {
	}
	if(DownloadIntegration && "shouldPersistDownload" in DownloadIntegration) {
		const bakKey = "_customButtonsDontRemoveFinishedDownloads_shouldPersistDownload";
		if(!patch ^ bakKey in DownloadIntegration)
			return;
		//LOG("dontRemoveFinishedDownloads(" + patch + ")");
		var store = "_store" in DownloadIntegration && DownloadIntegration._store;
		if(patch) {
			DownloadIntegration[bakKey] = DownloadIntegration.shouldPersistDownload;
			var wrapped = DownloadIntegration.shouldPersistDownload = function downloadPanelTweakerWrapper(download) {
				if(download.hasPartialData || !download.stopped)
					return true;
				//var retentionDays = prefs.get("downloadsMaxRetentionDays");
				return retentionDays > 0
					&& download.startTime > (Date.now() - retentionDays*24*60*60*1000);
			};
			if(store) {
				//LOG("dontRemoveFinishedDownloads(" + patch + "): override DownloadStore.onsaveitem");
				store[bakKey] = store.onsaveitem;
				store.onsaveitem = wrapped;
			}
		}
		else {
			DownloadIntegration.shouldPersistDownload = DownloadIntegration[bakKey];
			delete DownloadIntegration[bakKey];
			if(store && bakKey in store) {
				//LOG("dontRemoveFinishedDownloads(" + patch + "): restore DownloadStore.onsaveitem");
				store.onsaveitem = store[bakKey];
				delete store[bakKey];
			}
		}
	}
	else {
		dontRemoveFinishedDownloadsLegacy(patch);
	}
}
function dontRemoveFinishedDownloadsLegacy(patch) {
	const bakKey = "_customButtonsDontRemoveFinishedDownloads_downloads";
	const newKey = "_customButtonsDontRemoveFinishedDownloads_downloadsWrapper";
	if(!patch ^ bakKey in Services)
		return;
	//var logPrefix = "dontRemoveFinishedDownloadsLegacy(" + patch + "): ";
	if(patch) {
		var cleanUp = function downloadPanelTweakerWrapper() {
			var stack = new Error().stack;
			//LOG("Services.downloads.cleanUp()\n" + stack);
			if(
				stack.indexOf("@resource://app/components/DownloadsStartup.js:") != -1
				|| stack.indexOf("@resource://gre/components/DownloadsStartup.js:") != -1 // Firefox 20 and older
			) {
				//LOG("Prevent Services.downloads.cleanUp()");
				return undefined;
			}
			return downloads.cleanUp.apply(downloads, arguments);
		};
		if(newKey in Services) {
			var downloads = Services[newKey].__proto__;
			Services[bakKey] = null;
			setProperty(Services[newKey], "cleanUp", cleanUp);
			//LOG(logPrefix + "Will use old wrapper for Services.downloads");
		}
		else {
			var downloads = Services[bakKey] = Services.downloads;
			var downloadsWrapper = Services[newKey] = {
				__proto__: downloads,
				cleanUp: cleanUp
			};
			setProperty(Services, "downloads", downloadsWrapper);
			//LOG(logPrefix + "Create wrapper for Services.downloads");
		}
	}
	else {
		if(Services.downloads == Services[newKey] && Services[bakKey]) {
			setProperty(Services, "downloads", Services[bakKey]);
			delete Services[newKey];
			//LOG(logPrefix + "Restore Services.downloads");
		}
		else {
			// Yes, we create some memory leaks here, but it's better than break other extensions
			delete Services[newKey].cleanUp;
			//LOG(logPrefix + "Can't completely restore Services.downloads: detected third-party wrapper");
		}
		delete Services[bakKey];
	}
}
function setProperty(o, p, v) {
	Object.defineProperty(o, p, {
		value: v,
		configurable: true,
		writable: true,
		enumerable: true
	});
}

function destructor(reason) {
	if(reason == "update" || reason == "delete")
		dontRemoveFinishedDownloads(false);
}
if(
	typeof addDestructor == "function" // Custom Buttons 0.0.5.6pre4+
	&& addDestructor != ("addDestructor" in window && window.addDestructor)
)
	addDestructor(destructor, this);
else
	this.onDestroy = destructor;

dontRemoveFinishedDownloads(true);