Do I understand correctly that charities report contributions to the government, which then adjusts the taxes due from the donors? That seems like a good system.
The tax relief goes to the charity itself, rather than the donor. They can claim a top-up of up to 25% of your donation from the government, as long as the total amount claimed by all donation recipients does not exceed the total annual income tax paid by the donor. So if you donate £100 and fill out a declaration saying that you 'Gift Aid' that donation, the charity can claim an additional £25 from the government, as long as you paid at least £25 in income taxes that financial year.
Note that if you're paying in the 40% and/or 45% income tax brackets, the charity doesn't know about that, and still only gets the 25% topup (equivalent to the 20% income tax most people pay).
You can either:
* Gift that remaining amount to HMRC by doing nothing (it will not go to the charity)
* Claim it back (I don't know any other way to do this other than filing a self assessment)
I did not know until now that the other 20% could be reclaimed by the donor as well, if they paid 40% tax.
If the donor donates that amount as well and we loop infinitely, the final amount received by the charity would be exactly 1.5625x the originally-donated amount (assuming that the HMRC allows arbitrary small fractions of a penny to be claimed).
Explanation: if you donate an amount x, the final amount would be the infinite series x + sigma(0.25*0.2^(k-1) + 0.2^k, k=1, k=inf), which is the sum of two separate geometric series, so for each one, we can use the formula a/(1-r), where 'a' is the first term (0.25 for the first series, 0.2 for the second) and 'r' is the ratio between each term (0.2 in both cases).