What if the mempool exceeds 300 MB? The bitcoin mempool size is at 108 MB and still climbing according to chain.so. It is slowly approaching 300 megabytes, which is said to be the mempool limit for BTC. What happens after 300 MB?
Each node maintains a separate, individual mempool. While the default limits its size to 300 MiB, each node operator can set their own value. Note that the mempool size limit does not pertain to the serialized data or the transaction weight of the queued transactions (which is what is written to the block and what you see listed on a mempool monitors respectively), but to the actual memory usage of the deserialized transaction data on the node. This memory usage is platform dependent. In result, transactions start getting dropped when mempool monitors reach a block depth of about 80-95 blocks.
When a node's mempool limit is reached, it will drop the transactions with the lowest feerate and increase its minMempoolFeeRate. It will communicate its new minMempoolFeeRate to its peers, basically telling peers not to forward transactions below that feerate for the time being. Note that every node does this individually, so a node with a larger mempool or different architecture may drop transactions earlier or later. Nodes will keep a copy of transactions that are relevant to their own wallet. Even if all other nodes have dropped a transaction, the sender and recipients of a transaction would retain a copy. The sender can either force his node to drop the original transaction and send another conflicting transaction to update it, or their node will continue to try to broadcast the transaction so that it will eventually get relayed on the network again once the backlog has receded.
After the backlog decreases under the 300 MB limit and with some added delay, a node will lower their minMempoolFeeRate and start accepting transactions again that it previously expelled.
You can currently observe this happening e.g. on Johoe's mempool monitor¹: 
As you can see, the 1-2 sat/B band dropped around 7am that morning when the node's mempool exceeded its limit.

¹Note that there are two BTC views and the linked and relevant in this case is BTC (default mempool))