It might depend on the domain name space - but in general your options are extremely limited.
What typically happens is that if a domain name is not renewed it goes into a "holding" state during which only the registrar of record can renew the domain name - which means that they control most of the cards here.
If the domain name is not renewed after a stipulated period (3 months is common, but this may vary depending on the suffix of the domain - ie what tld/cctld it is) then the domain name is released back into a pool and anyone can register it. If the domain name is sought after, it can very often be snapped up by someone other then the original owner (how likely this is depends on the domain name and its history). There are services out there which can help you try pick up the domain name as soon as it falls off the registry. (This service is known by different names - domain drop catching, domain sniping. https://www.spamzilla.io/drop-catching-domains/ lists a number of providers that specialise in this)
Its a long shot, but depending on the suffix of the domain name and who the registrar is you MIGHT have some recourse -
A lot - probably most registrars are actually resellers of a registrar. If you registered the domain name through a reseller (which is likely - large registrars don't play these kinds of silly buggers - the risk to them is to high) - and you are very lucky - and very nice - and you can sufficiently prove ownership of the domain name you might occasionally be able to get the actual registrar to transfer the domain name to their registry directly and allow you to renew it.
Most TLD/CCTLD's have a dispute resolution process. This can be expensive and take a long time to resolve but may get you recourse to your domain name. It may be that the threat of dispute resolution might give you a bit of leverage as it might be that the registrar will be penalized by the entity that controls the TLD/CCTLD.
(A TLD is a generic suffix like .com .net .kiwi. A CCTLD is a country code tld - like .uk, .nz, .ru)