I think you can just turn up and pay but:
1. They will charge you full fare!
2. You may have to wait a while until they have space on a train.
Even if you can only book a week in advance you will get a far better deal!!
I'd also add that whilst going over the volume is spread over a few days. On the way back, most of it is Sunday night so you might have to wait a little longer if you haven't booked.
Tony