This problem would likely be solved prior to the value of 1 satoshi being $1

I would guess that one of the below solutions would be implemented when the value of one Bitcoin reaches either five or six figures when measured in USD.

1. The most likely solution would be that the Blockchain would be hardforked with a change to the Bitcoin protocol so that one bitcoin is divisible by (probably) 14 digits (however this value, x, could be anything as long as it is greater then 8.

2. The next most likely solution would be to have more off chain transactions with sites like Coinbase. Generally speaking, only larger transactions would be included in the blockchain so that he tx fee would be small compared to the size of funds being transferred.

This presents a number of issues. The primary issue would be this would cause centralization of bitcoins to major players, which is something that Bitcoin is suppose to stop/reverse. Another major issue would be that by the time this would happen the block rewards would likely be substantially lower then they are now (probably either 3.125 or 6.25 BTC plus tx fees per block) and having a lower amount of TX fees could compromise the security of the network as there would be less of an incentive to mine making it easier to execute double spend attacks.