Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous geometry on Wikipedia. It can take non-integer values; for example, the entire interval $[0,1]$ can be the range of a dimension function.
Also the Hausdorff dimension has a continuous range and is defined on the lattice of the subsets of a metric space. Is there any relation between these two concepts of dimension?