Say I host a web page on my computer, and you visit that site from somewhere in Europe. The web server on my computer will know your IP address, and I can choose to "forget" it as soon as the request is completed (eg. pipe apache logging to /dev/null) or keep them and abide by the GDPR such as responding to SAR's.
However there are a number of entities involved in the process that may, and probably do, keep logs of the transaction. These entities will include my ISP, BT (who provide the to house infrastructure), some number of backbone companies, and your ISP. I cannot access these logs, and given that I think it is a legal requirement to keep logs I think I cannot avoid such logging by choosing a different ISP.
How much of this logging am I responsible for under the GDPR?