I managed to open accounts with Chase while my SSN application was still pending / didn't have the paperwork in hand yet. They chased me up afterwards for it though and have sent me paperwork repeatedly asking for the information (SSN, reason for being in the US etc) in relation to any interest paid on accounts.

Is it possible to hold an account without any sort of tax ID number (ITIN or SSN) Surely one of them is required?

I've had a bank account and associated credit card with HSBC in the US for about 5 years now, and I only got an ITIN about 6 months ago. I managed to get the account because they offer a service for Premier customers where you can link accounts in different countries together, and I already had an account in the UK.

However, over the years it's been a massive PITA to maintain since I'm not a resident, and it's getting progressively more difficult - in fact I just moved all my money out of the US to a USD account in a more friendly jurisdiction (New Zealand, where I currently live) because it was getting to the stage where I could see them possibly freezing my account and it being a pain to sort out. I suspect I wouldn't be able to open the account under the same circumstances now.

Harris Bank allows Canadians to open checking accounts online. I don't have an SSN but I do have a US checking account that way.

It's easy to do it through American subsidiaries of Canadian banks like Harris. Not so easy for US-only banks and, as far as I know, impossible for virtual US banks.

I did this with Chase last year. Had to sign a mountain of paperwork though.

yup, bank of america took a lot of heat for doing that, mainly catering to immigrants that come into the US and want a bank account but don't always have an SSN...

Which bank? USAA refuses to do it.