Contemporary composers. Please help.

You must add Gyorgi Ligeti to you list. He does have some really far out stuff but also some very easy to like stuff as well. I listen to all the contemporary classical music and he is my favorite.

You might also check out Morton Feldman, some consider it to much repetition but since you like Reich I think you should give it a try.

One more mention that might easily be missed but I think is also one of the best is Giacinto Scelsi.

Thank you Ejlif,
Any particular Ligeti I should start with?

I remember starting with Ligeti Edition 7 chamber music, still one of my favorites. Ligeti Edition 3 works for piano is probably my all time favorite. I was able to see this performed live and that was a real treat (it took a world class pianist a year to learn to play these pieces) Also as far as larger orchestral works try out the Ligeti Project volume 2. These could be considered a little more difficult but once you get Ligeti a little bit and like it you can't not like these too. The Ligeti Project recordings are very good sounding too. If you really liked his stuff and you got all the Editions 1-7 and Project discs you would have it mostly covered.

Try Gorecki especially his Symphony #3. Dawn Upshaw's performance gets a lot of deserved play but my favorite is with Zolia Kilanowicz and the Polish National Orchestra.

Ludovico Einaudi may be more pop but his Divenire CD is a favorite as is Jim Chappelle's Dusk for solo piano

Take a look (listen) at Tan Dun as well.