Sorting this category is mostly a vocabulary problem. Five arrangements all get called free, and four of them let you network for a year without paying.
A permanent plan with a stated cap. Mesh at 1,000 connections, Contacts+ at one connected account and 1,000 records, HubSpot at 1,000 records and two users, Attio at three seats and 50,000 records, Cloze at contact management plus an email inbox. The cap is what you pay instead of money. Work out how many months it buys you.
A free tier of a product sold to sales teams. HubSpot and Attio are generous because free is the top of a funnel that ends in seats. That's a fair trade if you want a full CRM. It does mean the shape of the tool is deals, not relationships.
A countdown dressed as a plan. Dex, folk, Covve, Nimble, and Streak. Seven to fourteen days, then a bill.
Free if you run the server. Monica is open source and self-hosting costs nothing but your own hardware and your own patching. Your hosting provider may still want money.
Usage limits instead of a cap. The fifth arrangement meters volume rather than how many people you are allowed to know. Happenstance is free. The free plan is the whole product, with usage limits on chat and web searches beyond your network. Pro raises the limits: unlimited web searches and results, higher chat limits, and CSV export.