Large rupture strain (LRS) fiber-reinforced polymer (FRP) composites, typically formed from polyethylene naphthalate (PEN) and polyethylene terephthalate (PET) fibers, generally exhibit ultimate rupture strains >5%. Such fibers are particularly suited to the confinement of concrete columns on account of their LRS and sufficient elastic modulus. There are currently a limited number of studies on LRS FRP-confined concrete, particularly with high- and ultrahigh-strength concrete, so their behavior across a range of variables is still unknown. To improve this understanding, this paper systematically investigates the influence of fiber type, fiber thickness, and concrete strength on the behavior of FRP-confined concrete. To achieve this objective, the current investigation presents the results of 66 circular FRP-confined cylinders that are loaded concentrically. Three main parameters are investigated, namely, fiber type (i.e., PEN, PET, carbon, glass, and aramid), concrete strength (i.e., normal, high, and ultrahigh strength), and fiber thickness. The results show that regardless of fiber type, the stress–strain response is bilinear when the concrete is sufficiently confined. However, when there is insufficient confinement provided to the concrete core, the stress–strain response becomes trilinear. This trilinear response is more pronounced for LRS FRP-confined specimens because the confinement stiffness of the LRS FRP jacket is lower than that of a traditional FRP-confined specimen with an equivalent confinement ratio. Increasing the confining hoop stiffness (i.e., increasing FRP layers) reduces the magnitude of strength reduction after initial concrete cracking. It is also evident that as the unconfined concrete strength increases, the minimum confinement stiffness ratio necessary to prevent strength reduction after initial concrete cracking increases.