Question

Can a physical quantity have dimension but no unit

Answer 1

There is no physical quantity ,which has dimension but no unit.
Because the dimensional formula is based on seven fundamental physical quantities.They are  length ,mass,time,electric current,temperature,amount of substance,luminous intensity.
 And every fundamental physical quantity has a unit.
Eg; if we take the DF of Force - [MLT*-2]
                                       Unit is kgm/s*2           
So if a physical quantity has a dimension then it definitely has a unit.