Social media network Ello is currently receiving up to 31,000 requests an hour from people wishing to join its platform, its founder has told the BBC.
It was initially designed to just be used by about 90 friends of its founder Paul Budnitz.
But the bike shop owner, from the US state of Vermont, opened it to others on 7 August.
It has been dubbed the "anti-Facebook" network because of a pledge to carry no adverts or sell user data.
However some experts have cautioned that it might struggle with plans to charge micro-payments for certain "features".
The site has a minimalist design and does not appear as user-friendly, at first glance, as more established networks.
It has already survived a reported Distributed Denial of Service (DDoS) attack - a targeted flood of internet traffic - which briefly knocked it offline over the weekend.
"We're learning as we go but we have a very strong tech crew and back end," founder Paul Budnitz told the BBC.
"It's in beta and it's buggy and it does weird stuff - and it's all being fixed as quickly as we can."
Mr Budnitz added that he was "flattered" by the "anti-Facebook" description, but said that was not the way he saw his service.
"We don't consider Facebook to be a competitor. We see it as an ad platform and we are a network," he explained.
The network will eventually make money by selling access to features, Mr Budnitz added.
"Like the app store, we're going to sell features for a few dollars," he said.
Members can already check out features in development on the page and register their interest.
However, the traditional model of a free-to-use network has historically been the key to success, said James McQuivey, an analyst at tech research firm Forrester.
"Over all the other social media experiences from Whatsapp to Instagram to Pinterest - the reason they work is because they're free," he told the BBC.
"You don't invite your friend to connect with you if it costs your friend money. Even in the world of digital music - you can pay for services but most people don't."
"Ello is walking into a habit which consumers already have about digital services that they can't change on their own."
Mr McQuivey also suggested that people's attitudes towards advertising and data mining might not be as negative as they seem.
"We may all think we don't like advertising, we may believe we think it's wrong for companies to profit from our personal data but our behaviour suggests these companies give us what we want and we don't mind what they do in return," he said."
"The fact is nobody has ever made a significant move away from any internet provider because of advertising or data."