Multidimensional Arrays

;-)

Super-important to avoid combination of while and nested for-loops if not required, as makes the code more complicated than needed to be. Check out in the forum how other collegues are building up their solutions.

function solve(arr) {
    let account = arr.shift().split(' ');
    // let firstCurrentCommand = arr[0];
    // firstCurrentCommand = firstCurrentCommand.split(' ');
    // let currentCommand = arr.shift();
    // let newArr = [];
    // newArr.push(firstCurrentCommand);

    // while (currentCommand !== "Play!") {
    while (true) {
        // currentCommand = currentCommand.split(" ");
        let [command, game] = arr.shift().split(' ');

        if (command === 'Play!') {
            break;
        }
        // for (let i = 0; i < arr.length; i++) {
        // for (let j = 0; j < firstCurrentCommand.length; j++) {

        // if (currentCommand[i] === "Install") {
        if (command === "Install") {
            if (!account.includes(game)) {
                account.push(game);
            }
            // break;
        }

        // if (currentCommand[i] === "Uninstall") {
        else if (command === "Uninstall") {
            // if (currentCommand[i + 1] === firstCurrentCommand[j]) {
            if (account.includes(game)) {
                let gameIndex = account.indexOf(game);
                account.splice(gameIndex, 1);
                // firstCurrentCommand.splice(j, 1);
                // break;
            }
        }
        // if (currentCommand[i] === "Update") {
        else if (command === "Update") {
            // if (currentCommand[i + 1] === firstCurrentCommand[j]) {
            if (account.includes(game)) {
                let gameIndex = account.indexOf(game);
                account.splice(gameIndex, 1);
                account.push(game);
                // newArr.push(firstCurrentCommand[j]);
                // firstCurrentCommand.splice(j, 1);
                // break;
            }
        }
        // if (currentCommand[i] === "Expansion") {
        else if (command === "Expansion") {
            // firstCurrentCommand.push(currentCommand[i + 1].split('-').join(':'));
            // break;
            let originalGame = game.split('-')[0];
            if (account.includes(originalGame)) {
                let expansionGame = game.split('-')[0] + ':' + game.split('-')[1];
                let gameIndex = account.indexOf(originalGame);
                account.splice(gameIndex + 1, 0, expansionGame)
            }
        }
        // }

        // }
        // currentCommand = arr.shift();

    }
    console.log(account.join(' '));

}

Browser History (this the next step you should be going for in your solutions):

function solve(object, stringArray) {
    let newObj = {
        'Browser Name': object['Browser Name'],
        'Open Tabs': [...object['Open Tabs']],
        'Recently Closed': [...object['Recently Closed']],
        'Browser Logs': [...object['Browser Logs']],
    };

    for (let command of stringArray) {
        let data = command.split(' ');
        let currentCommand = data[0];
        let site = data[1];

        if (currentCommand === 'Open') {
            openTab(object, site);
        } else if (currentCommand === 'Close') {
            close(object, site);
        } else if (command === 'Clear History and Cache') {
            deleteObject(object);
        }
    }

    console.log(object['Browser Name']);
    console.log(`Open Tabs: ${object['Open Tabs'].join(', ')}`);
    console.log(`Recently Closed: ${object['Recently Closed'].join(', ')}`);
    console.log(`Browser Logs: ${object['Browser Logs'].join(', ')}`);

    function openTab(obj, site) {
        obj['Open Tabs'].push(site);
        browserLogs(object, 'Open ' + site);
    }

    function browserLogs(obj, command) {
        obj['Browser Logs'].push(command)
    }

    function close(obj, site, secondObj) {
        if (obj['Open Tabs'].includes(site)) {
            let index = obj['Open Tabs'].findIndex(x => x === site);
            let result = obj['Open Tabs'].splice(index, 1);
            obj['Recently Closed'].push(result[0]);
            browserLogs(object, 'Close ' + site);
        }
    }

    function deleteObject(obj) {
        obj['Open Tabs'] = [];
        obj['Recently Closed'] = [];
        obj['Browser Logs'] = [];
    }
}

solve({
    "Browser Name": "Google Chrome",
    "Open Tabs": ["Facebook", "YouTube", "Google Translate"],
    "Recently Closed": ["Yahoo", "Gmail"],
    "Browser Logs": ["Open YouTube", "Open Yahoo", "Open Google Translate", "Close Yahoo", "Open Gmail", "Close Gmail", "Open Facebook"]
},

    ["Close Facebook", "Open StackOverFlow", "Open Google"]);

solve({
    "Browser Name": "Mozilla Firefox",
    "Open Tabs": ["YouTube"],
    "Recently Closed": ["Gmail", "Dropbox"],
    "Browser Logs": ["Open Gmail", "Close Gmail", "Open Dropbox", "Open YouTube", "Close Dropbox"]
},
    ["Open Wikipedia", "Clear History and Cache", "Open Twitter"]);