  Tensor hierarchy algebras are infinite-dimensional generalisations of
Cartan-type Lie superalgebras. They are not contragredient, exhibiting an
asymmetry between positive and negative levels. These superalgebras have been a
focus of attention due to the fundamental role they play for extended geometry.
In the present paper, we examine tensor hierarchy algebras which are
super-extensions of hyperbolic Kac-Moody algebras. They contain novel algebraic
structures. Of particular interest is the extension of a hyperbolic algebra by
its fundamental module, an extension that contains and generalises the
extension of an affine Kac-Moody algebra by a Virasoro derivation $L_1$. A
conjecture about the complete superalgebra is formulated, relating it to the
corresponding Borcherds superalgebra.
