An example of an "unsupported browser with unsupported add-ons" is Firefox 3.6 running Google Toolbar. "Unsupported" means the software writers are no longer providing any updates to fix security flaws that are discovered.
The biggest risk is picking something up from browsing a website that serves up malicious code that your browser is vulnerable to. Don't think this is limited to the seedier side of the net as hacked advertising servers are a prime source - a lot of websites source advertising from third parties so have no control over the content.
The simple act of visiting a website can be enough. The only way to minimise the risk is to use the latest browser version.
Anti-virus software provides no guarantee against browser exploits.
FF3.6 currently has no known exploits that haven't been fixed but it's only a matter of time before it happens. Once an exploit is know it will be used maliciously and you'll be vulnerable.