  We study the maximal values of Betti numbers of tropical subvarieties of a
given dimension and degree in $\mathbb{TP}^n$. We provide a lower estimate for
the maximal value of the top Betti number, which naturally depends on the
dimension and degree, but also on the codimension. In particular, when the
codimension is large enough, this lower estimate is larger than the maximal
value of the corresponding Hodge number of complex algebraic projective
varieties of the given dimension and degree. In the case of surfaces, we extend
our study to all tropical homology groups. As a special case, we prove that
there exist planar tropical cubic curves of genus $g$ for any non-negative
integer $g$.
