Not a country, but the City of Santa Monica, California, keeps a registry of rents for all rent-controlled units. Landlords are required to report an initial tenancy, which sets the starting rent. Each year after that, the City tells the landlord what the "Maximum Allowed Rent" is. All the documents relating to each property are publicly available and the city publishes an annual report.

Not all cities with rent control keep such a registry. (e.g., Los Angeles doesn't keep a registry, but does tell landlords the maximum they can raise the rent each year.)