Set sequence and interval, paste in web console and run.

2048KeystrokeSim.js

var intervalID;
var left = 37;
var up = 38;
var right = 39;
var down = 40;
var keyboardEvent = document.createEvent("KeyboardEvent");
var initMethod = typeof keyboardEvent.initKeyboardEvent !== 'undefined' ? "initKeyboardEvent" : "initKeyEvent";
var i = 0;

// set interval of execution and sequence of keystrokes
var interval = 500;
var sequence = [up, left, down, right];

// clear the previous run
window.clearInterval(intervalID);

// set interval of execution
intervalID = window.setInterval(function () {
     var keyCodeArg = sequence[i];
     keyboardEvent[initMethod](
          "keydown", // event type : keydown, keyup, keypress
          true, // bubbles
          true, // cancelable
          window, // viewArg: should be window
          false, // ctrlKeyArg
          false, // altKeyArg
          false, // shiftKeyArg
          false, // metaKeyArg
          keyCodeArg, // keyCodeArg : unsigned long the virtual key code, else 0
          0 // charCodeArgs : unsigned long the Unicode character associated with the depressed key, else 0
     );
     document.dispatchEvent(keyboardEvent);
     if (i < sequence.length - 1) {
          i = i + 1;
     } else {
          i = 0;
     }
}, interval);