An infinite-type surface $Sigma$ is of type $mathcal{S}$ if it has an isolated puncture $p$ and admits shift maps. This includes all infinite-type surfaces with an isolated puncture outside of two sporadic classes. Given such a surface, we construct an infinite family of intrinsically infinite-type mapping classes that act loxodromically on the relative arc graph $mathcal{A}(Sigma, p)$. J. Bavard produced such an element for the plane minus a Cantor set, and our result gives the first examples of such mapping classes for all other surfaces of type $mathcal{S}$. The elements we construct are the composition of three shift maps on $Sigma$, and we give an alternate characterization of these elements as a composition of a pseudo-Anosov on a finite-type subsurface of $Sigma$ and a standard shift map. We then explicitly find their limit points on the boundary of $mathcal{A}(Sigma,p)$ and their limiting geodesic laminations. Finally, we show that these infinite-type elements can be used to prove that Map$(Sigma,p)$ has an infinite-dimensional space of quasimorphisms.
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