Yep. The extra second is the 60th second. http://earthsky.org/human-world/leap-second-june-30-december...

There's a bunch of ways the leap second can be reflected. :59 twice, :60, leap second smearing (adding small amounts of time leading up to midnight). How your particular system reacts depends on the use, how it's configured, etc.

That makes sense. I was being slightly facetious, and I think most devices would NTP re-sync after some interval. But it would be cool if my phone did in fact show 3:59:60 (I'm pacific time) and/or analog smartwatch paused on the 59 for two seconds.