How to check the depth of an object?

Well, here you go buddy, a function that does exactly what you need!

utils.depthOf = function(object) {
    var level = 1;
    for(var key in object) {
        if (!object.hasOwnProperty(key)) continue;

        if(typeof object[key] == 'object'){
            var depth = utils.depthOf(object[key]) + 1;
            level = Math.max(depth, level);
        }
    }
    return level;
}

A lot easier than we thought it would be. The issue was how it was incremented, it shouldn't have been recursively adding, rather getting the bottom-most and adding one, then choosing the max between two siblings.

Back from the dead! Throwing my solution into the mix -

function depth (t, mem = new Set)
{ if (mem.has(t))
    return Infinity
  else switch (mem.add(t), t?.constructor)
  { case Object:
    case Array:
      return 1 + Math.max
       ( -1
       , ...Object
           .values(t)
           .map(_ => depth(_, mem))
       )
    default:
      return 0
  }
}

console.log(depth({a: {b: {c: "z"}}}))   // 3
console.log(depth({a: "z"}))             // 1
console.log(depth({}))                   // 0
console.log(depth("z"))                  // 0
console.log(depth({a: [{b: "z"}]}))      // 3

const a = []
a[0] = a
console.log(depth(a))                    // Infinity

This old question was recently resurrected and I don't see any answers as simple as this one (to be fair, this uses techniques not available when the question was written):

const objectDepth = (o) =>
  Object (o) === o ? 1 + Math .max (-1, ... Object .values(o) .map (objectDepth)) : 0

console .log (objectDepth ({foo: {bar: {baz: 'baa'}}}))
console .log (objectDepth ({abc: 'xyz'}))

This, like most answers here, will fail when the input object is cyclic. An answer that addresses that limitation would require much more sophistication.