…clearing a paper jam in the copier.
I’m the office champion. :).
What’s your skill which can never be automated or outsourced?
It’s not my skill, but a robot will never be able to play the guitar with soul.
Coaching of business managers.
Mentoring.
Consulting.
Psycho-therapy.
cleaning dirt off of little kids’ faces
Mathematical research. It always starts with a question and I cannot imagine a computer “wondering”.
“Noticing” something from what’s right there & “assigning” meaning to the perception.
I can injure a human being or, through inaction, allow a human being to come to harm.
I can disobey orders given me by human beings even where disobeying such orders could injure a human being or, through inaction, allow a human being to come to harm.
I don’t have to protect my own existence, or I can protect my own existence even if such protection could injure a human being or, through inaction, allow a human being to come to harm, or be the result of disobeying orders given to me by a human being.
Given sufficiently advanced robots, none. Unless you define robot in such a way that precludes being capable to do some thing, such as “A robot is a thing that cannot feel emotions”.
Cooking okra or grits properly.
Screw up in the epic ways that I can. 
I’m pretty adept at being an asshole. Robots aren’t going to pull that off.
Mom-Bot 9000’s actuators whirred softly as its RealFace[sup]TM[/sup] angled downward towards the boy. Camera-eyes fed into an image interpretation software package that quickly recognized the smudges and categorized them as needing prompt - but not emergency - attention. Tiny motors and rotors turned with infinite precision as Mom-Bot picked up a dish cloth from the counter. Nozzles in its “mouth” spritzed water onto its “tongue”, which extended towards to the cloth and licked it several times. Speech software ran a subroutine to check the language setting, which returned the value “English (American, Contemporary)”, which set off a millisecond cascade resulting in Mom-Bot’s speakers intoning: “C’mere - Let me get that for you.”
The boy froze in horror as the rag was rubbed across his face.
whine?
guess if I feel like having sex
avoid cooking dinner
the list could go on and on
Masturbating.
I’ll say art. Anyone can tell the difference between some photobashed piece of concept art and something created by a human from scratch.
People pay good money for paint randomly smeared on canvas buy monkeys (sometimes literally). I’m sure a robot can do that.
I bet a robot will never be able to take the last cup of coffee in the break room then pretend there’s enough left for someone else as an excuse to avoid making another pot.
I’m going to say that I think you’re already wrong about this.
Iamus, a music composition computer system who’s works have been performed by the London Symphony Orchestra, among others.
eDavid, the painting robot has a very impressive gallery.
Amazon has been selling books written by computers for years now.
I built a robot in my garage. I programmed it so that:
it could go down to the quarry and throw stuff in there
Without human assistance, can eDavid make something like this? It looks like you feed him a complex input and he spits an image out. eDavid has no imagination.
The real answer is of course virtually nothing, like Frodo said. But that doesn’t make for a very fun topic.
I don’t forsee many robo-clergy out there saving souls. Maybe in the year 3000 when the robot devil is out there taking hands and stuff, but not in our times.