Is it possible for the registrar to increase the fee after a year if I purchase and promote a domain name within that time frame? Additionally, do I have any special rights to the domain name once my original purchase expires?

I would like to clarify my ownership of the domain - if I invest heavily in promotion, can the registrar suddenly increase the price due to high demand or popularity of the domain? Given that domain names can be purchased online with only an email address, this is a valid concern.

When you "buy" a domain name, you're actually renting it with the opportunity to renew it once the rental period is up. If you're using a Standard com, net, or country domain such as RU, US, or GB, then there's no need to be overly worried. You can simply transfer your domain to another registrar if the current one raises their prices too high.
but, things may be a little different for national zones like FR or US, as local laws could potentially result in your domain being taken away if it infringes upon someone else's intellectual property rights or company name.

If there is no legally binding agreement between a registrar and a client, it is almost guaranteed to result in a scam. Often, when a website on a rented domain starts to become successful in commercial areas, the registrar will think that they can charge more for it. They may not consider how much effort and resources the client invested into developing that website and domain, such as content creation, link building, and optimization.

The registrar may simply trigger the thought process of "this is MY domain, and it should cost more" and either ask an unrealistic amount for rent or sell it to someone else. Therefore, if you have your own website and are creating a valuable resource for someone else, it's not an ideal situation. Unfortunately, this is something I have learned from experience.

It's sad, but we don't buy, we rent a domain name. You have already noticed that a certain price only applies to one or three or five years, then the price increases by 25 or 50% (this is a common practice). At the legislative level (I will not give the source, because everything depends on the registrar) it is stated that the price cannot exceed 135% of the transaction amount.

If the contract of the offer says that the amount of the renewal of the contract may exceed this amount and you signed it (few people read the contracts), then of course you can expect the domain name to grow significantly. From scratch, the price cannot rise in a year. Read the contract.

It is indeed possible for some registrars to have clauses in their terms of service that allow them to raise the renewal fees after a certain period, especially if they see increased demand or popularity for a particular domain name.
When you invest heavily in promoting a domain name, you are essentially adding value to it through your marketing efforts. This can attract more attention and potentially drive up the demand for the domain. In such cases, registrars may see an opportunity to capitalize on the situation by raising the renewal fees, taking advantage of the domain's newfound popularity.

In terms of special rights to the domain name once your original purchase expires, it largely depends on the registrar's policies and the specific terms under which you initially acquired the domain. Some registrars offer automatic renewal options, ensuring that you retain ownership of the domain as long as the renewal fees are paid. However, if the fees are raised significantly, this could create a financial burden for you, especially if you have invested heavily in promotion.

This brings into question the ethics of domain name pricing and ownership. It raises the issue of whether registrars should be allowed to substantially increase fees for existing domain owners based on external factors such as demand and popularity. It also underscores the need for transparency and fairness in domain name transactions, as well as the importance of protecting the interests of domain investors and promoters.

While there may not be a straightforward answer to your concerns, it's important to carefully review the terms of service of the registrar you choose and consider the potential implications of fee changes before making a significant investment in promoting a domain name. Additionally, staying informed about industry developments and engaging in dialogue with registrars can help shape fair and equitable practices in the domain name ecosystem.