i.e. Is there a finite number of transactions that will ever take place using Bitcoin?
I have Googled this somewhat and found discussions about Bitcoin's limited, finite supply, but it's unclear to me whether the blockchain itself is a finite resource.
No, there is no limit on the number of blocks in blockchain. The transaction can go on for ever.
The limit is only on the bitcoin that will ever be produced in the environment (Why was 21 million picked as the number of bitcoins to be created?) , but no limit on the number of times it can be exchanged.
The finite amount of Bitcoin (21,000,000) has little to do with the number of blocks in the blockchain. Although the miners (the people who create the blocks) are rewarded with newly generated Bitcoin which will get fewer and fewer by time, they will still create blocks after they get no newly generated Bitcoin at all. Why? Because they get the fees in addition. In the far future all miners will be rewarded with the fees only and still generate blocks.
