Compressive strength is one of the most critical mechanical properties of various types of concrete and is the main input variable for structural concrete design. Recently, with the advances in concrete technology, applications of fiber-reinforced concrete (FRC), including ultra-high performance concrete (UHPC), have grown rapidly. These new types of concrete are well known to exhibit superior mechanical characteristics such as compressive strength, fracture toughness, and durability compared to conventional concrete and thus are popularly used in urgent repair jobs where compressive strength is an important parameter to determine the required curing time until open to the public. Considering the importance of compressive strength in practice, this study aims to evaluate the effect of age and maturity on the compressive strength characteristics of three different types of concrete, namely UHPC with micro and macro steel fibers, FRC, and plain concrete, and to propose a new design strength criterion for UHPC. To this end, 180 concrete cube specimens were tested at 12 different ages between 3 and 126 days. The results indicated that irrespective of the type and presence of fibers, UHPC gained more than 90% of their ultimate compressive strength after only 21 days, while FRC and plain concrete specimens required a longer time (i.e., 28 days) to achieve 90% of their ultimate strength. Therefore, UHPC may adopt a 21-day compressive strength as a design input instead of a 28-day compressive strength commonly required for structural concrete specified by many codes of practice. Moreover, the obtained experimental results were compared with existing compressive strength predictive models in the codes of practice.