A space-intensive solution...

circular_skipping.js

// N number of people stand in a circle. 
// The people are numbered in order from 1 to N. 
// Starting from 1, we remove every other person 
// from the circle until there is only one person remaining. 
// I would like you to write a function which takes in the number N 
// and outputs the number of the last person remaining. For example:

// f(3) outputs 3
// f(4) outputs 1
// f(5) outputs 3

var f = function(N) {
  var circle = Array.apply(null, Array(N))
    .map(function(item, index) { return index + 1; });
  var cursor = 0,
      removing = 0;
  while (circle.length > 2) {
    circle.splice(cursor, removing);
    cursor = (cursor + +!removing) % (circle.length);
    removing = +!removing;
  }
  return circle[(cursor + 2) % (circle.length)];
};

// TESTS
console.log(f(3))
console.log(f(4))
console.log(f(5))