Someone should ask Magnus Carlsen.

Edit: fortunately the fine article brings this up and goes further to point out that go is fundamental to self improvement in some cultures and will likely have a long life after AlphaGo.

[13th century] In the age of gunpowder, do young archery prodigies have a future?

This is a good one. It was still many many centuries later before it stopped being a tool in war and from a competitive prospect it is still kickin (although not pervasive)

In the age of Watson, does Jeopardy have a future?

Did Jeopardy ever really have a purpose?

Jeopardy has many purposes. It provides employment, entertainment, and a cultural touchstone, as well as a benchmark of our quest to replicate our nature. Relatively few things have more purpose.

In the age of high tech robot replacements, do unskilled factory workers have a future?

This one unfortunately isn't analogous with the others.

They actually do not.

The analogy only works if people do the particular activity for pleasure, so the question becomes... is it possible to find pleasure in factory work? One could argue that it's possible, but it may look different to what it does today. Some people may enjoy the feeling of refined manual dexterity, so games could emerge where activities like packing boxes at speed becomes a sport, in a similar vein to how cup stacking is a sport now.