// This solution stores the reverse order in the nodes themselves. It's hacky so I like it :)

LinkedList.prototype.reverse = function () { var q;

  // Make doubly linked
  for (q = this.head; q; q = q.next ) {
    if ( q.next ) q.next.prev = q; else this.head = q;
  }

  // Reverse  
  for (q = this.head; q; q = q.prev ) {
    q.next = q.prev;
  }
  
  // Better clean up
  for (q = this.head; q; q = q.next ) {
    q.prev = undefined;
  }

};

http://jsbin.com/ugojoq/43/edit

[EDIT]

Ah, I didn't know, but it possible to reverse a singly linked list without remembering the entire list:

  var p = this.head, q, r;
  while ( p ) {
    r = q;
    q = p;
    p = p.next;
    q.next = r;
  }
  this.head = q;

(blatantly based on an answer found on StackOverflow).

http://jsbin.com/ugojoq/67/edit

TIL :) [/EDIT]