We define and study new classifications of qcb0-spaces based on the idea to measure the complexity of their bases. The new classifications complement those given by the hierarchies of qcb0-spaces introduced in [7,8] and provide new tools to investigate non-countably based qcb0-spaces. As a by-product, we show that there is no universal qcb0- space and establish several apparently new properties of the Kleene- Kreisel continuous functionals of countable types.