  Smoktunowicz, Lenagan, and the second-named author recently gave an example
of a nil algebra of Gelfand-Kirillov dimension at most three. Their
construction requires a countable base field, however. We show that for any
field $k$ and any monotonically increasing function $f(n)$ which grows
super-polynomially but subexponentially there exists an infinite-dimensional
finitely generated nil $k$-algebra whose growth is asymptotically bounded by
$f(n)$. This construction gives the first examples of nil algebras of
subexponential growth over uncountable fields.
