The purpose of this research is to investigate and characterize the time-dependent behavior of new concrete compositions made by replacing cement with secondary aggregates — small clay particles — and then comparing them with control specimens made from standard concrete. Cement is one of the main constituents of standard concrete, but the carbon dioxide released into the atmosphere during the cement production process contributes to the global warming and the development of holes in the ozone layer. The use of clay particles in concrete production can make the construction industry more environmentally friendly. This paper presents experimental test results of the compression creep of the proposed concretes. Several concrete compositions with unconventional additives were designed and prepared. The tests were performed on both normal strength and high strength concretes. Specimens were tested in two extreme environments: in one case there was 100% humidity provided by protecting the specimens from desiccation, and in the other case specimens were air-dried and protected from any moisture. Concrete specimens were subject to load in a constant room temperature and with a constant level of moisture. The investigated properties include compression strength, modulus of elasticity and creep. The creep test results showed that after approximately 90 days of loading, the response seemed to be smaller for high strength concretes than for normal strength concrete.