This is not a good practice  and I am not recommending it, but sure it can be done: Live demo (click).
//list each file name you want to find
var files = ['some-file', 'some-other-file'];
//list possible extensions
var exts = ['html', 'doc'];
//iterate each file name
$.each(files, function(index, file) {
  //attempt to get the file
  //pass a copy of the "exts" array so that each function can "shift" through it
  getFile(file, exts.slice());
});
function getFile(file, exts) {
  //the filename to try (added extension)
  var newFile = file+'.'+exts.shift();
  //try to get the file
  $.get(newFile).then(function(data) {
    //found the file - do something with the contents
    foo(data);
  }, function() {
    //file not found
    //if there are any extensions left to try
    if (exts.length) {
      //try again (will shift to next extension)
      getFile(file, exts);
    }
  });
}
function foo(data) {
  var div = document.createElement('div');
  div.textContent = data;
  document.body.appendChild(div);
}