I recently had to get a new HDD so I transferred my only Bitcoin to Blockchain so I can get it off later. Although, coming back to it now, I have absolutely blanked on the password for it. I understand that they don't reset passwords, but this is really confusing me.
It isn't that they
don't reset passwords. It's that they
can't reset passwords. They don't have access to your current password. It isn't stored on their servers. They have no way of knowing what it is. What they have is an encrypted copy of your wallet (just like the backups you have). To change the password, they would have to first decrypt the wallet with the password that you don't have, and then re-encrypt it with the new password. Obviously if they don't have the current password and you don't have the current password, then this isn't possible.
All my passwords are variants of a single base depending on what I'm using the password for, so naturally I used my 20 character financial password to first log in. This didn't work, so I ended up going through all my passwords that I have ever used for anything. None of them worked.
Assuming that you actually used a variant of your "single base" I can think of a couple of possibilities.
It's possible that you simply used a variant that you haven't thought of yet. If you continue to think about it, and try additional variants, you might come up with the right password. It might also be possible to write a program that tries all possible variations of your single base
It's also possible that you made a typo when initially entering your password. You can try various possible common typos and see if any of them result in the correct password. It might also be possible to write a program that tries all possible typos on all possible variations of your single base.
Depending on how complex your password is, such programs may be able to find the password in a few hours, or it may take years. If your password is really complex with a lot of likely variations, it might not be possible to find it at all.
I have that Blockchain account sync'd with Authy on a jailbroken iPhone, is there anything I could do there?
I'm not familiar with Authy. If it stores a copy of your password, then I suppose there might be a possibility there.