We report a detailed low-temperature thermodynamic investigation (heat capacity and magnetization) of the superconducting state of KFe 2As2 for H||c axis. Our measurements reveal that the properties of KFe2As2 are dominated by a relatively large nodeless energy gap (δ0 = 1.9 κBTc) which excludes dx2-y2 symmetry. We prove the existence of several additional extremely small gaps (δ0 < 1.0 k BTc) that have a profound impact on the low-temperature and lowfield behavior, similar to MgB2, CeCoIn5, and PrOs4Sb12. The zero-field heat capacity is analyzed in a realistic selfconsistent 4-band BCS model which reproduces only qualitatively the recent laser ARPES results of Okazaki et al. [Science 337, 1314 (2012)]. Our results show that extremely low-temperature measurements, i.e., T < 0.1 K, are required in order to resolve the question of the existence of line nodes in this compound.