Trivia games with baby name to celebrity matching, baby trivia, etc. and baby shower bingo during gift opening are the least annoying games, in my experience.
Bingo is the only one that does not make me roll my eyes and I enjoy it at bridal and baby showers...playing along makes watching the gifts even more fun.
Re: Non cheesy baby shower games
The only game that I have heard of that didn't make me cringe was maybe doing celebrity baby name trivia.
We didn't do games at mine-- my hostess assigned every guest a number before hand and she had me pull numbers out of a hat to hand out prize
BFP 1- EDD 2/09/11 Missed MC DX @11 weeks D&C- 7/25/10 BFP 2- EDD 12/22/11 Natural MC @ 5w 2d BFP 3- EDD 1/25/12 DD Josephine born 1/16/12