This is basically a combination. A "traditional" chess program would use a tree search, but trees get quickly ot of hand since they grow exponentially. The trick is to prune them, and they trained a network to do that. It selects just the moves that look good to it. (It has some level of randomness to it, too) After reaching deep enough in the search tree, they use another network to evaluate who's winning. Usually this is hard to do in Go, and that's why the second network is quite novel and helpful.

So, they use a combination of techniques. And they're doing well at it.

right, yes, but my question was meant to be a bit more general - this and various other results have shown that it is possible to train a deep net to do a specific task very successfully - my question was if it's possible to train it to do two or more tasks as successfully or will the network then have to be exponentially larger. I suppose there is no known way to "combine" trained networks together.

The standard is to have each neural net be trained just for one task, as you say; there may be research into multi-skill neural nets but I have yet to see any. AlphaGo in particular is extremely specialized to Go, even in terms of how the algorithm is implemented.