I have two-step verification set up so that it sends a text to my phone, I can login to this discussion base through a web browser and have no problem receiving a verification code to my phone to do that.
As for signing into icloud on my laptop I don't receive a verification code once it prompts that one has been sent, I have waited up to 30 minutes for one and nothing arrives.
The laptop is currently running Version 10.8.5 due to the restoring process, I cannot update to a newer version without logging into icloud. It is the system where you type your password followed by the six digits into the password field.
Since I don't receive a code I simply hit enter again with a random 6 digit number since I never received one. Then my phone (I do not own an iphone so it sends via text) will receive either one or two verification codes. I will then put in the most recent verification code, it will state it is wrong and then I receive two more verification codes at the exact same time.
And to clarify I have to sign in to icloud in order to access the app store where I can get system updates.
And when I get two verification codes simultaneously I will either receive them from the same number or two different apple numbers, never knowing which one is the most recent. It appears that sending the verification code doesn't get triggered until I type in a random number first because just hitting enter with only the password just prompts the same response of to enter password followed by verification code. I have also tried going with the less recent verification code when I receive double but nothing works it always says it wrong and continues to send two more codes each time.
I have attempted multiple different approaches to see if something would work and have waited 24hours between each attempt to allow time for the codes to balance out again.