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Most of the molecules detected thus far in exoplanet atmospheres, such as water and CO, are present for a large range of pressures and temperatures. In contrast, metal hydrides exist in much more specific regimes of parameter space, and so can be used as probes of atmospheric conditions. Iron hydride (FeH) is a dominant source of opacity in low-mass stars and brown dwarfs, and evidence for its existence in exoplanets has recently been observed at low resolution. We performed a systematic search of archival CARMENES near-infrared data for signatures of FeH during transits of 12 exoplanets. These planets span a large range of equilibrium temperatures (600 $\lesssim T_{eq} \lesssim$ 4000K) and surface gravities (2.5 $\lesssim \mathrm{log} g \lesssim$ 3.5). We did not find a statistically significant FeH signal in any of the atmospheres, but obtained potential low-confidence signals (SNR$\sim$3) in two planets, WASP-33b and MASCARA-2b. Previous modeling of exoplanet atmospheres indicate that the highest volume mixing ratios (VMRs) of 10$^{-7}$ to 10$^{-9}$ are expected for temperatures between 1800 and 3000K and log $g \gtrsim3$. The two planets for which we find low-confidence signals are in the regime where strong FeH absorption is expected. We performed injection and recovery tests for each planet and determined that FeH would be detected in every planet for VMRs $\geq 10^{-6}$, and could be detected in some planets for VMRs as low as 10$^{-9.5}$. Additional observations are necessary to conclusively detect FeH and assess its role in the temperature structures of hot Jupiter atmospheres.