Wouldn't somebody somewhere in this kind of transaction have to pay the VAT, thus rendering this impossible?

The premise itself is fiction. The P in GDP doesn't stand for exchange or transaction. It stands for "Product." It measures the net amount of new wealth, not how many times existing wealth is transferred among parties. If I sing you a song and you pay me $1 million dollars and this process continues back and forth, the net amount of value when we stop is still $1 million and the net change to GDP is zero.

People indulge a lot of strange and wrong ideas about economics and, not surprisingly, those bad ideas are typically well aligned with their preferred world views.

Services are also part of GDP. If I pay you to sing, that's part of GDP.

That's true, but it doesn't salvage your argument. Your accounting fiction isn't a 'service' for the purposes of GDP.

Yes it is. It's a pretty common criticism of GDP.

The other comment had it right though - taxes would kill this scheme.

"It's a pretty common criticism of GDP."

...with no more veracity than your hypothetical. You don't understand what GDP measures. You've allowed yourself to believe that the economists involved are naive. You are correct, however, that this sort of bad thinking is common. Very common.

If I find a shiny rock and sell it to you for $100 and you find a fossil and sell it to me for $100, that's $200 added to the GDP. If we trade rather than pay each other, then it's $0 added to GDP. The same goes for services. If I mow your lawn, paint your house, hypnotize you to quit smoking, or sing you a song - all of those things (are included in GDP.

When you pay for entertainment it's 100% included in GDP the same way as when you pay for somebody to inspect your car.