You're going to have tons of variety and options in Asia, it being the biggest continent and housing more than half of the people...
My favorites from my travels:
Mongolia--lucked out that my visit coincided with the Dalai Lama's. Starkly Beautiful. Mmmm horse.
Turkey--especially Cappadocia and Ephesus, oh, and get the Turkish Massage. It's an experience you'll never forget, no matter how hard you try.
Vietnam--I was just around Ho Chi Min City. Fascinating.
India--think of it as all the diversity of Europe in one country.
Russia--Lake Baikal, the pearl of Siberia. Russians. Gotta love 'em.
Israel--for an exploration of Western Religion, it's hard to beat.
Japan--Beautiful. Attention to detail.
And the non-highlights:
China and Hong Kong--Mainly just for my personal tastes and (probably bad) choices while there. Plus, China was a little harder to navigate back when I was there. Probably better now.