I was asked this question on another list. After a little poking around, I was surprised to learn that there were a few still in existence, albeit with some restrictions. I'm going to pass on the information so that you can tell someone who needs dialup Internet access (such as on that older computer running Linux that you just gave them for free) but can't afford a monthly subscription service.
NOTE: There is one more ISP, JUNO/NETZERO, which offers 10 hours a month of Internet service for free, and has lots of local numbers nationwide, but they make you look at lots of ads and have a Windows-only client (and most people report that they add up your time so sneakily you rarely get 10 hours a month).
All of the options below are Linux-friendly free dialup ISPs.
If you live in Maine, Massachusetts, New Hampshire, New Jersey, New York, Rhode Island or Southern Florida, *OR* if you have unlimited long-distance service on your phone line, there is a regional ISP which offers free local dialup service (and thus free dialup service outside their region if your long-distance is unlimited).
http://www.freedialup.org/site/
You register on their website to find out their list of phone numbers, then dial into a server phone number. The username is "your phone number", the password can be anything at all (please pick a non-offensive word).
The service has two limitations: the ports for SMTP (outgoing E-mail) and NNTP (usenet newsgroups) have been blocked, so you will not be able to do either one, even with a third-party service. All other services, including POP3, are available. It is suggested that you use web-based E-mail to do E-mail with their service.
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The Western Washington State region, including Seattle, has its own free dialup ISP, and like the East Coast free dialup ISP they don't mind if you call long-distance (with your unlimited long-distance service) to use their ISP:
http://www.nocharge.com/connect.htm
Like the East Coast free dialup ISP, the SMTP (outgoing E-mail) port is blocked on their service, but otherwise the Internet service is unfiltered. You can use their Windows Dialer or your own dialup connection software (thus this one is also good for Macintosh and Linux users). The username and password are right there on the page, and you don't even have to register on their website to use their free dialup ISP.
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If the problem is that you only use the Internet occasionally and can't afford a monthly plan, but might occasionally want to be able to use more than 10 hours of Internet in a single month, "Budget Dialup" is a company which offers prepaid Internet service. On a Windows system, you can use their software and get the first five hours free, but you don't have to use their software to connect to their dialup servers, meaning you can use other operating systems like Mac OS or Linux.
You buy a package of time, and then have a year in which to use it. The current plans, for calling a number local to you, are 10 hours for $5.95, 20 hours for $9.95, and 60 hours for $28.95. If you don't do a lot of downloading, 20 hours will last you for a month or two of browsing and E-mail. Buying additional time will give you one more year to use both the new time and any old time from previous purchases.