Is there anything god can't do?

Show up & exist.

He can't stop being God, I guess. 

God cannot lie and He cannot deny Himself.  There are things that God COULD do but won't - especially just for your amusement.  Do you think God is a puppet on a string?  The devil - through that thief on the Cross, asked Jesus to get them all off the cross if He was REALLY GOD.  Well, He IS REALLY GOD.  That must have been quite tempting considering the terrible agony and pain and suffering He was experiencing, but He didn't give in did H?  If He had, nobody would be saved today.  So think yourselves lucky that you have a very powerful God who doesn't just do anything on a whim.

The idea of a supreme infallible all powerful being does create some paradoxes. One of the more famous is the question could god create a rock so heavy that even he can't lift it. I asked a similar question in a more humorous way a few weeks ago but someone reported it. See meme below, but it is a similar question that points out such paradox.

Yes. He cannot lie. He cannot forget. He cannot hate any human being. He cannot do anything evil.

Most answers here show that people don't know the difference between "CAN'T" and "WON'T". And it's most of the atheists, who don't believe any God exists, who pontificate smugly on what they claim God can't do. The simple answer is that God will not do anything contrary to His perfect nature, otherwise He would not be God. God must be true to Himself. And He is.

Hebrews 6:18 says: "in order that through two unchangeable things in which it is IMPOSSIBLE FOR GOD TO LIE, we who have fled to the refuge may have strong encouragement to take firm hold of the hope set before us.

He can't be good and evil at the same time.

No, but he acts within his timetable and accordance to his will.

The answer to your question is a qualified no. But your question is a loaded question. There are many things God will not do.

Yes, he can't break his promises.