I have now heard back from the agent. As predicted he didn't have a solution beyond what I have already learned here.
He did say that the number attached to the texts was 90007 but I tried texting stop all to it and it would not send. I noted that this is the last part of the longer number 30090007 which appears to be Tap2Bill so I sent the same text to that. It did send but I have had no response at all.
They didn't answer the phone and haven't responded to the message I left. I seem to be running out of options.
I'm afraid this is yet another example of a clueless GiffGaff agent. The number 30090007 is used by Tap2Bill for sending Payforit receipts. It is not the five digit number related to the subscription. Sending STOP to 30090007 or 90007 won't help. With this kind of advice from GiffGaff agents, it's not surprising that members become exasperated when dealing with Payforit scams!
It does, however, confirm that you need to talk to Tap2Bill to find out who has been charging you. It also suggests that your earlier text to 83463 will probbly have been successful in stopping the charges.
Tap2Bill should be able to tell you who was taking your money, so that you can demand a FULL refund.
The 0333 003 0599 number will probably only be active during UK working hours, so try again tomorrow. You can also email them at customer.service<at>tap2bill.com.
Thanks again, @muggles708. I shall try again to speak to them tomorrow. It does look as though I'm on my own as far as GG is concerned. Thank goodness for knowledgeable people like you!
It's a good idea to get something in writing to confirm that you have had contact with the third party company which took your money. Then, if you are not satisfied with the outcome, you can "escalate" the matter to GiffGaff in accordance with the Payforit rules.
A failure by GiffGaff to meet their obligations could be cause for a formal complaint which could ultimately go to the ombudsman.