We prove existence results concerning equations of the type $-\Gd_pu=P(u)+\gm$ for $p>1$ and $F_k[-u]=P(u)+\gm$ with $1\leq k<\frac{N}{2}$ in a bounded domain $\Omega$ or the whole $\mathbb{R}^N$, where $\gm$ is a positive Radon measure and $P(u)\sim e^{au^\beta}$ with $a>0$ and $\beta\geq 1$. Sufficient conditions for existence are expressed in terms of the fractional maximal potential of $\gm$. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of $\gm$. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form $u={\bf W}_{\alpha,p}^R[P(u)]+f$ in $\mathbb{R}^N$, where $0