What was the happiest moment for Rei?
When she's holding Shinji's hands.
That one time Shinji cleaned her room.
That's it, unless you wanna go outside the anime.
Rei isn't a very happy girl.
With Gendo's hand inside her tit.
What does Rei like?
And the time Shinji pulled her out of the Eva and told her to smile.
I guess she was happy.
Being Rei is suffering.
Being appreciated as a person.
When Shinji opened her entry plug and told her to smile she was "happy." I'd say that she at least had a sense of purpose and was somewhat happy because of it when she sacrificed herself to kill the angel infecting her eva.