Sorry if I am missing something but doesn't EOS meet this description?
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Nope.
First of all, it has a token which serves as a currency, no matter what people might call it.
EOS is not fully decentralized. It does not let anybody fully participate. It is not fully censorship resistant. You are at the mercy of 21 Block producers.
It is also an oligarchy, in which the people who have more are able to decide more. One of the main reasons of having a non-monetary blockchain is to prevent exactly that.
At best, you can say that EOS is a private blockchain with a public outlet.
I personally think EOS is junk. It is the antithesis of what blockchain is supposed to be.
I was a big proponent of the DPoS system as a general concept, but looking at Lisk, I see big problems in the political part of the system. I still like projects like SHIFT and Adamant also looks kinda interesting, but they may face similar problems once they get more popular.
As for the others, they all still have some token/coin with a value pegged to them. I have the feeling that this token/coin hinders adoption in multiple ways.
a) It makes it hard to interact with the blockchain. the_zhang_wei* is on to something here: if you have to buy BTC, then go to some shady exchange, get the token and then can interact with a system which, for the average user, is slower and clunckier than other systems, nobody is going to use it.
b) The fact that your coin might be worth double of what it is worth today makes users hesitant to use it. While speculation may be a great tool to get people interested and a great thing to have for systems focussing on being actual money, it is a big problem for utility tokens and coupon style solutions.
A non-monetary blockchain would still need some type of input, maybe even PoW, so users will have to pay somehow for the service they are using, but the fact that this is not connected to a tangible token, which can be traded, shuts off a whole slew of problems you would normally face.
I can imagine a hybrid version, in which a backbone-chain is maintained by a monetary system (or something existing, like ETH or BTC is used), but the main action happens on a non-monetary system.
Thread reference:
https://bitcointalk.org/index.php?topic=4100004.0