We present the results of a Spitzer/Herschel infrared photometric analysis of the largest (716) and highest-redshift (z = 1.8) sample of Brightest Cluster Galaxies (BCGs), those from the Spitzer Adaptation of the Red-Sequence Cluster Survey (SpARCS). Given the tension that exists between model predictions and recent observations of BCGs at z < 2, we aim to uncover the dominant physical mechanism(s) guiding the stellar mass buildup of this special class of galaxies, the most massive in the Universe and uniquely residing at the centres of galaxy clusters. Through a comparison of their stacked, broadband, infrared spectral energy distributions (SEDs) to a variety of SED model templates in the literature, we identify the major sources of their infrared energy output, in multiple redshift bins between 0 < z < 1.8. We derive estimates of various BCG physical parameters from the stacked νL ν SEDs, from which we infer a star-forming, as opposed to a 'red and dead' population of galaxies, producing tens to hundreds of solar masses per year down to z = 0.5. This discovery challenges the accepted belief that BCGs should only passively evolve through a series of gas-poor, minor mergers since z ∼ 4 (De Lucia & Blaizot 2007), but agrees with the improved semi-analytic model of hierarchical structure formation of Tonini et al. (2012), which predicts star-forming BCGs throughout the epoch considered. We attribute the star formation inferred from the stacked infrared SEDs to both major and minor 'wet' (gas-rich) mergers, based on a lack of key signatures (to date) of cooling-flow-induced star-formation, as well as a number of observational and simulation-based studies that support this scenario.