Fully private and proprietary blockchains
There is no mainstream application of these types of blockchains, as they deviate from the core concept of decentralization in blockchain technology. Nonetheless, in specific private settings within an organization, there could be a need to share data and provide some level of guarantee of the authenticity of the data.
An example of this type of blockchain might be to allow for collaboration and the sharing data between various government departments. In that case, no complex consensus mechanism is required, apart from simple state machine replication and an agreement protocol with known central validators. Even in private blockchains, tokens are not really required, but they can be used as means of transferring value or representing some real-world asset.