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In this work we study global boundedness and exponential integrability of weak solutions to degenerate $p$-Poisson equations using an iterative method of De Giorgi type. Given a symmetric, non-negative definite matrix valued function $Q$ defined on a bounded domain $Ω\Subset\mathbb{R}^n$, a weight function $v\in L^1_\textrm{loc}(Ω,dx)$, and a suitable non-negative function $τ$, we give sufficient conditions for any weak solution to the Dirichlet problem \begin{align*} \begin{array}{rccl} -\displaystyle\frac{1}{v}\mathrm{div}\left(\left|\sqrt{Q}\nabla u\right|^{p-2}Q\nabla u\right)+τ\left|u\right|^{p-2}u&=&f&\textrm{in }Ω, \end{array} \end{align*} \begin{align*} \begin{array}{rccl} u&= & 0&\textrm{on }\partialΩ \end{array} \end{align*} to be bounded and exponentially integrable when the data function $f$ belongs to an appropriate Orlicz space.