Compositionality beyond bases

. I motivate introducing this framework by showing that existing definitions of bases in linguistics and in the study of notation diverge significantly from each other, making neither suitable for a more general discussion of numeration systems across all representational formats. While notational definitions lump too many systems under the 'not-a-base' label, linguistics-inspired approaches allow too many systems into the base kingdom. I illustrate the broad range of compositional tools that need to be included in a format-independent discussion of numeration systems by showing how perceptual grouping in tallies instantiates shifts in counting units and how place-value can implement a recursive unit-labelling function. I highlight the advantages of an anchor-based framework by applying it to some well-known lexical systems.This article is part of the theme issue 'A solid base for scaling up: the structure of numeration systems'.