This is the Bitcoin board so I'll be speaking about the Bitcoin blockchain....

The first and most evident "limitation' of blockchains is scalability.
Public blockchains are very hard to scale especially on-chain.
This is because of the way bitcoin achieves consensus; most nodes must verify a transaction (or block) is valid.
If the block size is too high then a lot of nodes will be cut off and this leaves the power of the network in the hands of a few people which leads to centralisation.
Also, if the block size is too low then obviously a lot of transactions won't be able to fit and this leads to high fees during high thoroughput periods as people will pay more for their transaction to be mined.