Leaky paywalls for SaaS products

Has anyone every tried, or even thought through, a leaky paywall for a SaaS product? A leaky paywall is like what you experience on the New York Times website. You get to use the product a few times before it prompts you to sign up and pay to continue using it. It's different to freemium where you have to first at least sign up. The pro of a leaky paywall is not having to sign up to see the value in the product. The con is they are very easy to get around for extended use and tracking of use doesn't go across devices, profiles etc. You also don't get any details of your "free" users even. No email to push for conversion.