A lot of people look for domain names, which cost nothing. Unfortunately, however, there is no such thing as free domains, as there ultimately is a catch somewhere.
If you would be getting the services of a web hosting service provider that offers a free domain name as well, then it is certain that your web pages would contain a lot of advertising, which you would have no control over and would not bring you any income at all.
Also, there could be several limitations and restrictions, which could make you decide that free domains are not really free after all.
Another downside of making use of free domains is that you would most likely end up having a domain that would just be a sub-domain instead of an actual domain name. So your domain name would be google.com/yourdomain or yourdomain.google.com. With a domain name like that, you would have a hard time establishing a good online business name that customers could trust and be confident in.
Creating and building and designing your own website requires a lot of dedication, effort and hard work so that you would be able to make sure that it would be generating a lot of visitors that could potentially turn into paying customers. What if you get a free domain name and then find your online business a success, which would give you the need to get a unique domain name? The traffic that you already have worked so hard to generate could just be wasted. This means that if you are planning to use the domain name for an online business, it would be a lot smarter to get a unique domain name which is all your own. Although, it would cost you more at the start, ultimately, you would realize that it is a good investment.
However, if you just set up a website for personal use or for a hobby or interest that you have, then a free domain would probably work just well for you.