The way Sareth manipulates Cleon XVII into ignoring all arguments against his idiotic (and eventually fatal) idea of going to Terminus in person. It's blatantly obvious to everyone else in the room that she is trying to get him out of the way, but Day is completely distracted by her use of sex appeal and leaning on his masculinity and ego.
Towards the end of their negotiation, Vault-AI Seldon near-openly talks past Cleon XVII to Demerzel, knowing both that Day is too unstable to seriously engage with and that she is the real power behind the Empire. Day once again doesn't even notice.
Hober Mallow taunting Cleon XVII as he returns to the Shining Destiny:
Oy, you gelded bastard! Yesterday! Temporary Emperor. [...] I'm sorry I couldn't stay longer when I came to Trantor. Just missed you.