Context. Globular clusters (GCs) are established emitters of high-energy (HE, 100 MeV < E < 100 GeV) gamma-ray radiation which could originate from the cumulative emission of the numerous millisecond pulsars (msPSRs) in the clusters' cores or from inverse Compton (IC) scattering of relativistic leptons accelerated in the GC environment. These stellar clusters could also constitute a new class of sources in the very-high-energy (VHE, E > 100 GeV) gamma-ray regime, judging from the recent detection of a signal from the direction of Terzan 5 with the H.E.S.S. telescope array. Aims. To search for VHE gamma-ray sources associated with other GCs, and to put constraints on leptonic emission models, we systematically analyzed the observations towards 15 GCs taken with the H. E. S. S. array of imaging atmospheric Cherenkov telescopes. Methods. We searched for point-like and extended VHE gamma-ray emission from each GC in our sample and also performed a stacking analysis combining the data from all GCs to investigate the hypothesis of a population of faint emitters. Assuming IC emission as the origin of the VHE gamma-ray signal from the direction of Terzan 5, we calculated the expected gamma-ray flux from each of the 15 GCs, based on their number of millisecond pulsars, their optical brightness and the energy density of background photon fields. Results. We did not detect significant VHE gamma-ray emission from any of the 15 GCs in either of the two analyses. Given the uncertainties related to the parameter determinations, the obtained flux upper limits allow to rule out the simple IC/msPSR scaling model for NGC6388 and NGC7078. The upper limits derived from the stacking analyses are factors between 2 and 50 below the flux predicted by the simple leptonic scaling model, depending on the assumed source extent and the dominant target photon fields. Therefore, Terzan 5 still remains exceptional among all GCs, as the VHE gamma-ray emission either arises from extra-ordinarily efficient leptonic processes, or from a recent catastrophic event, or is even unrelated to the GC itself.