  We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$
over arbitrary field, which gives rise to an associative hull $\mathbf A$, a
Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$,
$\mathbf K$. One has a natural filtration for $\mathbf A$ which associated
graded algebra has a structure of a Poisson superalgebra and $\mathrm{gr}
\mathbf A\cong\mathbf P$, also $\mathbf P$ admits an algebraic quantization.
The Lie superalgebra $\mathbf Q$ is finely $\mathbb Z^3$-graded by multidegree
in the generators, $\mathbf A$, $\mathbf P$ are $\mathbb Z^3$-graded, while
$\mathbf J$, $\mathbf K$ are $\mathbb{Z}^4$-graded. These five superalgebras
have clear monomial bases and slow polynomial growth. We describe
multihomogeneous coordinates of bases of $\mathbf Q$, $\mathbf A$, $\mathbf P$
in space as bounded by "almost cubic paraboloids". A similar hypersurface in
$\mathbb R^4$ bounds monomials of $\mathbf J$, $\mathbf K$. Constructions of
the paper can be applied to Lie superalgebras studied before and get Poisson
and Jordan superalgebras as well. The algebras ${\mathbf Q}$, ${\mathbf A}$,
and the algebras without unit $\mathbf P^o$, $\mathbf J^o$, $\mathbf K^o$ are
direct sums of two locally nilpotent subalgebras and there are continuum such
decompositions. Also, $\mathbf Q=\mathbf Q_{\bar 0}\oplus \mathbf Q_{\bar 1}$
is a nil graded Lie superalgebra. In case $\mathrm{char}\, K=2$, $\mathbf Q$
has a structure of a restricted Lie algebra with a nil $p$-mapping. The Jordan
superalgebra $\mathbf K$ is just infinite nil finely $\mathbb Z^4$-graded,
while such examples (say, analogues of the Grigorchuk group) of Lie and Jordan
algebras in characteristic zero do not exist. We call $\mathbf Q$, $\mathbf A$,
$\mathbf P$, $\mathbf J$, $\mathbf K$ fractal because they contain infinitely
many copies of themselves.
