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I suggest an unmarked paper bag.

So you're basically asking that GOG allow you, when you already have an identifying username, email, and probably GUID to basically wire money.

prepaid steam card

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Are you sure that using a bank account is safer to purchase online? Ypu can do it via PayPal, adding a bank account, is really safe and you can get back your money easily!

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Become a farmer, trade your produce for game codes. It's the only way.

What about PaySafe cards? You can buy them anonymously at any store and use them without giving away any data.

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Paypal works for me. They allow many options between using your own bank account, your own credit cards, or a Paypal balance.

Beyond that, Paysafe cards are properly anonymous. You can buy them with cash at a vendor, and spend them however you like. Otherwise, you're likely spinning your wheels since you're actually complicating the payment process for GOG or the Bank by forcing them to deal with transaction data they usually don't have to deal with (i.e. Routing numbers, ID verification, etc). There's a REASON why people pay intermediaries to handle the minutiae they don't want to deal with.

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If you're really concerned about big data, paysafecard is the best option. More anonymous than a bank transfer.

I care deeply about privacy. I use PaySafeCard.

Literally this - it's basically the same as doing a wire/BACS transfer. Although for various reasons, it's far more sensible to use a credit card to make payments than a debit card.

@OP - I'm a bit confused as to what you want. If you're interested in privacy, wiring cash directly is a really bad idea (your bank will know what you're spending cash on - which is no better than a credit card processing company knowing this). So, do you want a solution that offers privacy (in which case you've already got Paysafe cards), or do you want to avoid third parties to make payments (in which case, just use your debit card)