I will define an elementary theory of non-holomorphic modular forms and
describe some of its basic properties.
Within this family, there exists a class of functions which correspond
to certain mixed motives.
They are constructed out of single-valued iterated integrals of holomorphic
modular forms, and are closely related to a problem in string theory.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/7239