Q:

Code is executing multiple times per event: Multiple downloads

I want to download a JSON object from within the content script.  At first, when I request the download, it downloads one file, but at the second request, it downloads two files; at the third request, three files are downloaded, etc.  
Background.js
chrome.browserAction.onClicked.addListener(function (tab) {
    download();
    chrome.runtime.onMessage.addListener(function (request, sender, sendResponse) {
        //Alert the message
        console.log(request);
        chrome.downloads.download({
            url: request.method,
            filename: request.name
        }, function (downloadId) {

        });
        //You have to choose which part of the response you want to display 
        //  ie.  request.method
        //alert('The message from the content script: ' + request.method);
        //Construct & send a response
        sendResponse({
            response: "Message received"
        });
    });
});

function download() {
    chrome.tabs.executeScript(null, {
        file: "jquery.js"
    }, function () {
        chrome.tabs.executeScript(null, {
            file: "content_script.js"
        });
    });
}

Content Script
function sendMessage(url, filename) {
    //Construct & send message
    chrome.runtime.sendMessage({
        method: url,
        name: filename
    }, function (response) {
        //Alert the message
        //You have to choose which part of the response you want to display 
        //  ie.   response.response

        //alert("The response from the background page: " + response.response);
    });
}
var json = JSON.stringify(ticket);
var blob = new Blob([json], {
    type: "application/json"
});
var url = URL.createObjectURL(blob);
var a = document.createElement('a');
a.download = "backup.json";
a.href = url;
a.textContent = "Download backup.json";
var container = document.getElementById('ticketDetail');
//container.appendChild(a);
var fName = ticket.date.replace(".", "_") 
sendMessage(url, fName.replace(".", "_")  + ".json");

A:

As is usually the case with this pattern of problem, the issue is that you are adding multiple anonymous listeners to an event.  Specifically, you are adding yet another chrome.runtime.onMessage listener each time the action_button is clicked.  You need to add the listener only once. 
The simple solution to this is to just add the chrome.runtime.onMessage once:
chrome.browserAction.onClicked.addListener(function (tab) {
    download();
});

chrome.runtime.onMessage.addListener(function (request, sender, sendResponse) {
    //Alert the message
    console.log(request);
    chrome.downloads.download({
        url: request.method,
        filename: request.name
    }, function (downloadId) {
    });
    //You have to choose which part of the response you want to display 
    //  ie.  request.method
    //alert('The message from the content script: ' + request.method);
    //Construct & send a response
    sendResponse({
        response: "Message received"
    });
});

function download() {
    chrome.tabs.executeScript(null, {
        file: "jquery.js"
    }, function () {
        chrome.tabs.executeScript(null, {
            file: "content_script.js"
        });
    });
}

