how a system based on a blockchain would deal with exponential growth of informations carried in it? I was reading this article about the difference between a blockchain and centralized servers and a question came up to my mind: if I understand well the difference stands in the fact that in a blockchain there isn't a centralized server that maintains online a version of the informations based on it and protects them from corruption. I'm thinking for instance about wikipedia. Every "miner" instead has a copy of the informations - of the blockchain - and that is why it is so difficoult to alter a blockchain. 

But what if these informations, and with them the blockchain itself, became too heavy to be supported by little private servers? My question is how a system based on a blockchain deals with exponential growth of informations. 

I hope to ask in the right place, if it isn't I beg your pardon!
Bitcoin's blockchain grows currently at the rate of one block every ten minutes on average, according to consensus parameters. That's not an exponential growth, but a linear one. At present this is well below the growth in storage capacity allowed for by Moore's-law.

If in future the consensus parameters were to change, either to make blocks more frequent or, larger, then the storage requirements are one consideration that would have to be taken in hand.

In 2009 we had the first 2.0 terabyte hard drive, by early 2018 it is expected that a 14TB HDD should be available.[1] The cost per Gigabyte has reduced from nearly $0.12 per GB in 2009 almost down to $0.02 per GB in 2017.[2]

If the cost in whatever terms of running a full node becomes too onerous then we would expect to see more SPV nodes and, SPV nodes connecting to full node servers i.e. Electrum.


  "Bitcoin wallets like Electrum have been using SPV successfully for years without any problems, and they save users the hassles of maintaining a full network node."[3]


SPV was discussed in section 8 of the original Bitcoin whitepaper.

rel:
[1] Wikipedia - History of hard disk drives - https://en.wikipedia.org/wiki/History_of_hard_disk_drives  

[2] BackBlaze - Hard Drive Cost Per Gigabyte - https://www.backblaze.com/blog/hard-drive-cost-per-gigabyte/

[3] medium.com - Why Every Bitcoin User Should Understand “SPV Security” - https://medium.com/@jonaldfyookball/why-every-bitcoin-user-should-understand-spv-security-520d1d45e0b9