Go problems feel super compelling to me. Typical problems can be boring "counting" exercises, but I find it hard to beat something like the ear reddening move: http://senseis.xmp.net/?EarReddeningMove

There is a great series on youtube on fantastic moves from professional games with commentary in English: http://www.youtube.com/watch?v=CJ9Oexs59CE&feature=relmf...

They are really beautiful in part because they become tesuji through a confluence of factors that reverberate across the entire board. It can be hard to see for amateurs (including me) but once you give them the right context and insight, they become startlingly brilliant.

Also, it's interesting you find it hard to make a critical mistake in Go - this feels very common to me. For example, a decision like deciding to defend a group instead of sacrificing it (which comes up all the time) often snowballs really quickly.