  We introduce the algebra of formal multiple Eisenstein series and study its
derivations. This algebra is motivated by the classical multiple Eisenstein
series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein
series and multiple zeta values. In depth one, we obtain formal versions of the
Eisenstein series satisfying the same algebraic relations as the classical
Eisenstein series. In particular, they generate an algebra whose elements we
call formal quasimodular forms. We show that the algebra of formal multiple
Eisenstein series is an $\mathfrak{sl}_2$-algebra by formalizing the usual
derivations for quasimodular forms and extending them naturally to the whole
algebra. Additionally, we introduce some families of derivations for general
quasi-shuffle algebras, providing a broader context for these derivations.
Further, we prove that a quotient of this algebra is isomorphic to the algebra
of formal multiple zeta values. This gives a novel and purely formal approach
to classical (quasi)modular forms and builds a new link between (formal)
multiple zeta values and modular forms.
