Solutions of differential equations that blow up on the boundary of a domain are called \textit{large solutions} or \textit{boundary blow-up solutions}.

We investigate large solutions for the fractional Laplacian on a sequence of cylinders $ B_{1}^{n-1}(0)\times [-\ell,\ell]$ as $\ell\to +\infty$ and obtain large solutions on infinite cylinder $\Omega_{\infty}:=B_{1}^{n-1}(0)\times\mathbb{R}$ to the following problem
\[
\begin{cases}
(-\Delta)^{s} u = f(x,u) & \text{in } \Omega_{\infty},\\
u = g & \text{in } \mathbb{R}^{n} \setminus \Omega_{\infty},\\
\delta_{\Omega_{\infty}}^{s}(x)\,u(x) \xrightarrow[x\to \theta]{x\in \Omega_{\infty}} h(\theta) & \text{for } \theta \in \partial\Omega_{\infty},
\end{cases}
\]
under suitable assumptions on $\big(f, g,h)$. The investigation is carried out for both linear and semi-linear equations. Further, for a linear equation, we identify three distinct types of large solutions on an infinite cylinder. These are solutions arising from the data $(f,0,0)$, $(0,g,0)$, and $(0,0,h)$.