  In this paper, we propose a generalization of the Almansi decomposition for
slice regular functions of several quaternionic variables. Our method yields
$2^n$ distinct and unique decompositions for any slice function with domain in
$\mathbb{H}^n$. Depending on the choice of the decomposition, every component
is given explicitly, uniquely determined and exhibits desirable properties,
such as harmonicity and circularity in the selected variables. As consequences
of these decompositions, we give another proof of Fueter's Theorem in
$\mathbb{H}^n$, establish the biharmonicity of slice regular functions in every
variable and derive mean value and Poisson formulas for them.
