I saw your other comment that mentions your phone obeys 2Factor. Are you by any chance not running Ice Cream Sandwich? After I completely wipe my Galaxy Nexus it still doesn't ask for 2Factor. It doesn't even require an application specific password, it accepts my real password with no hesitation.
You've probably accidentally disabled the two-factor auth on your account. I strongly recommend checking.
My phone always asks for two-factor auth. In fact, I had to wipe my phone and re-auth, so I used one of the throw-away codes. When I re-init'd Google Auth app (which annoyingly requires disabling and reenabling two-factor auth, AND invalidates one time use keys), it immediately reprompted me to complete an oAuth cycle with the two-factor code for the core Google account on my phone.
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I saw your other comment that mentions your phone obeys 2Factor. Are you by any chance not running Ice Cream Sandwich? After I completely wipe my Galaxy Nexus it still doesn't ask for 2Factor. It doesn't even require an application specific password, it accepts my real password with no hesitation.
You've probably accidentally disabled the two-factor auth on your account. I strongly recommend checking.
My phone always asks for two-factor auth. In fact, I had to wipe my phone and re-auth, so I used one of the throw-away codes. When I re-init'd Google Auth app (which annoyingly requires disabling and reenabling two-factor auth, AND invalidates one time use keys), it immediately reprompted me to complete an oAuth cycle with the two-factor code for the core Google account on my phone.
ICS/Galaxy Nexus/etc
Well this is embarrassing. I checked and somehow it was disabled. Thank you.