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This is only partly true according to me. There is still the transaction fee which is paid out to the one, keeping the blockchain alive. Don't forget that there are blockchain available were all coins were already pre-mined and they also keep on running!
But indeed, I do wonder what will happen to the bitcoin blockchain once all coins are mined.

well, if you have fees it kind of means there is currency there? ;)

Yeah but I was referring to a network in which there is a cryptocurrency present!
Without crypto of course no mining, nor transaction fees (or at least not via the blockchain, it could be possible that the creator of the blockchain will ask a subscription fee, which would be payable outside the blockchain).