Question

give an example of each
a unitless quantity with dimension
a dimensionless quantity with units

Answer 1

dimension less quantity

      =  plane angle          units  degrees or  radians.  symbol  °
     = solid angle              units steradians  symbol   symbol  sr
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unit less quantity  ---  but dimension  exists for that quantity.

     I wonder if there is a quantity like that.  If there is a dimension, that means there is a unit and there is a way of measuring that.  That means there is no such quantity without a unit but with a dimension .