  We construct positive singular solutions for the problem $-\Delta u=\lambda
\exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$,
having a prescribed behaviour around the origin. Our study extends the one in
Y. Miyamoto [Y. Miyamoto, A limit equation and bifurcation diagrams of
semilinear elliptic equations with general supercritical growth. J.
Differential Equations \textbf{264} (2018), 2684--2707] for such
nonlinearities. Our approach is then carried out to elliptic equations
featuring iterated exponentials.
