Is it possible that transactions throughput will grow to beyond what can be synchronized? The number of users increases, and thus the number of transactions. Is it possible that in the future transactions arise faster than I can sync them?
When a transaction is broadcast to the network it enters the "mempool". Every full node maintains a pool of unconfirmed transactions in RAM memory. Newly broadcast transactions are relayed from node to node through the network.

A typical mining node selects, from its mempool, as many transactions (with attractive fees) as will fit in a block and then attempts to find a valid hash for that block. The first miner to find a valid hash for a block containing a particular transaction will then confirm that transaction.

Any and all transactions not selected for inclusion in a block remain in mempool. A node typically discards unconfirmed transactions from the mempool after 72 hours.

Hence, since memory is limited on most nodes, heavy transaction load can result in their mempool becoming full. This leads to congestion and a slow-down in transaction propagation (through the network) as well as delays in confirmation, since many transactions might not enter the mempool and will have to be rebroadcast by wallet clients until the congestion subsides.

Layer 2 solutions, such as Lightning Network, propose to allow almost unlimited, near-instant transactions per day, thereby mitigating the limited capacity of the Bitcoin network (Layer 1).