You need it for bank accounts, for benefits, for tax, for shopping, for taking cash out, for getting the lower prices at the supermarket, for booking doctor's appointments, paying for prescriptions, for getting test results, for work, for leisure.
Some of those would depend on whether you 'count' a third party using the internet to facilitate you using a manual service. E.g.
Withdrawing cash over the counter in the bank (where the bank will be using the internet to provide that service).
Shopping - if you count the shop's till using the internet.
Doctor's appointments - can book by phone but will go on online system.
Prescriptions - can get a paper one and pay cash, but see tills above.
How far do you take it? Say you said you'd only shop in markets where the cash gets shoved in a cash bag rather than put through a till - but somewhere in the supply chain and accounting chain, the internet will be involved.
To be completely free of the internet you'd probably have to be self-sufficient or join a self-sufficient commune.