This paper studies the coefficients of algebraic functions. First, we recall the too-little-known fact that these coefficients fn have a closed form. Then, we study their asymptotics, known to be of the type fn∼CAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents α can not be 1/3 or −5/2, they in fact belong to a subset of dyadic numbers. We extend what Philippe Flajolet called the Drmota-Lalley-Woods theorem (which is assuring α=−3/2 as soon as a "dependency graph" associated to the algebraic system defining the function is strongly connected): We fully characterize the possible critical exponents in the non-strongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not $\mathbb{N}-algebraic). We end by discussing some extensions of this work (limit laws, systems involving non-polynomial entire functions, algorithmic aspects).