In BS EN 12390-1: 2012, it states that "For each shape of test specimen, cube, cylinder and prism, the nominal size d (Figures 1, 2 and 3) should be chosen to be at least three and a half times the maximum aggregate size (Dmax according to EN 206-1) in the concrete." The minimum ratio of nominal size of test specimen to the maximum aggregate size is to ensure even distribution of stress in the specimen under compression test. In Section 5.5.1.1 of BS EN 206-1: 2013, strength tests shall be based on tests carried out on either 150/300 mm
cylinders or 150 mm cubes conforming to EN 12390-1. However, it also states that other sizes of moulded specimens may also be used provided the relationship to those standardized has been established and documented.
Cube moulds are used for testing of hardened cement concrete works. The purpose of testing hardened concrete is to confirm that the concrete used at site has developed the required strength.
Normally it is minimum requirement at site to provide evidence of cube test result to confirm the required concrete strength.
The cubes are normally tested on 7 and 28 days, until any early specific test is required. 6 Cubes are casted at the time of concreting, 3 cubes for 7 day testing and 3 cubes for 28 day testing. The cubes are removed from the curing tank and dried, then the cubes are tested by compressive testing machine (CTM) and values are recorded.
Usually at site the cube specimen used is of the size 150 x 150 x 150 mm, if the largest nominal aggregate size does not exceed 20mm.100 x 100 x 100 mm cubes may also be used as an alternative.
Although there are other sizes of cubes available and used for concrete testing such as 50mm, 225mm and 300mm. The Indian Standard : 10088-1992 specification for mould is used for concrete testing. Mild Steel(IS: 210-1975) and Cast Iron(IS:210-1978) is used for construction of cube moulds.
Size of concrete cube as per BIS recommandation is 150 mm, if compressive strength expected is too high and could not get crushing load with available compression testing machine, size has to reduced no option
If the test results are to be used for conformity check in a contract, you'll need to refer to the project specification to see if the cube size is specified without any flexibility allowed. In case reference to BS EN 12390 is allowed, you may make use the clause I previously suggested. Also, if the test results are solely used for research or product development purpose, I believe you can use 75 mm cubes in particular for ultra high strength concrete such that you can use a compression machine with lower crushing capacity.