Especially with the new explosions, the battles are great. If only I didn't have to almost micromanage my capital ships... then I could just sit back and watch. (i still need to get about to die ships out of there before they do, which requires me to watch their health bars)
If you recorded it then you can sit back and watch it!
My first favorite moment was when I first perchased SoaSE. I was TEC, playing on Systems of War against three others, one of whom was my ally. I watched my ally explore then go back when I finally had enough resources for more infrastructure maintainence. This one game, however, a fellow TEC A.I. was able to get the Distant Star Drive researched early and decided to ransake an asteroid colony of mine. I chased him off and when the pirate raids came around, I made sure he was targeted. Well, that made him angry, and so he came down upon me with most of his fleet. After harassing a better portion of my planets, he left, saying don't do it again or he'll destroy me. That was my favorite moment in that part because we actually talked, threatening one another and I believed he would do that if I did it again.
Later in that same game, having unleashed myself from my galaxy, I visited his and conquered it. But he didn't allow me to have his home planet so easily, and a HUGE battle between my fleet of Kodiaks and Light Frigates against his Carriers and Capital Ships was just awesome. I have several screenshots somewhere of it. It was my first big battle, and I knew then that this game was for me. All that slow economy buildup and slow plotting was worth the battles like that one.