It's not actually infinite.

The blockchain is located on each node's hard drive. Many people are running nodes using hard drives like the ones you have, so, if we let a large amount of messages get written on the blockchain, here node's will run out of disk space.

Also, real people who are trying to send money now have to wait longer for their transactions to confirm because of a bunch of message "transactions" in the mempool (unconfirmed transaction waiting list).