I'm wondering, if we see bitcoin, or just any cryptocurrency based upon the blockchain principle, large, whether the volume of the chain is not going to be problematic.

If we look at companies such as Visa, they processes several thousands of transactions per second.  Now let us assume that the minimum amount of data for a transaction is 100 bytes (send and receive address, and amount and some data) in a transaction-optimised set of cryptocurrencies.  Let us assume that 1000 transactions per second happen.  That means 100 KB per second on the block chain.  That is a data flux that is still reasonable to keep up with, but when accumulated, it is huge.  It is more than 8 GB per day.  The block chains of the cryptocurrencies taking these transactions would grow at a pace of 8 GB a day, which means 3.2 TB a year.

Of course, the burden can be distributed over different cryptos.  You only need to look at the block chain of the crypto you're interested in.  If there would be a few tens of cryptos around, sharing the burden, each chain would only grow with a few hundred GB a year.  But still.  These are big volumes, even in the future.