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We pay stuff that should be essentially free, but it is not limited to internet access only.
I don't have a free internet, I have to pay a monthly fee.

I also have to pay monthly fee but net neutrality will enable ISP's to make us pay even more money to visit certain sites.
It's gonna be worse
Right now you're paying for access to internet but then you will also have to pay for surfing it.

You mean to pay for website membership?

Not only that, Any big website will be blocked unless you pay for it like YouTube right now it's free but then you will have to pay for it.

Yes, I am aware what are you saying. Some news websites have that already. Like a big newspapers. You can only see like 1-2 paragraphs of the text and you have to pay for the rest.
This is a way they protect their journalist and make money, otherwise nobody would pay for their work.
To be absolutely honest, they can do it on the part of the network, like you said, the big websites, but it would be fairly unbelievable to impose the same paid model for the entire internet.
Take in consideration the decentralization, it just wouldn't work.