sed -i -e 's/Game #[0-9]*/&1/' file
-i is for in-place editing, and & means whatever matched from the pattern. If you don't want to overwrite the file, omit the -i flag.
-i
&
Using sed:
sed
cat file | sed -e 's/\(Game #[0-9]*\)/\11/'
sed 's/ Game #\([0-9]*\):/ Game #1\1:/' yourfile.txt