There is growing evidence that intermediate-mass black holes (IMBHs), defined here as having a mass in the range M=500-105 M, are present in the dense centers of certain globular clusters (GCs). Gravitational waves from their mergers with other IMBHs or with stellar BHs in the cluster are mostly emitted in frequencies 10 Hz, which unfortunately is out of reach for current ground-based observatories such as advanced LIGO (aLIGO). Nevertheless, we show that aLIGO measurements can be used to efficiently probe one of the possible formation mechanisms of IMBHs in GCs, namely a runaway merger process of stellar seed BHs. In this case, aLIGO will be sensitive to the lower-mass rungs of the merger ladder, ranging from the seed BH mass to masses 50-300 M, where the background from standard mergers is expected to be very low. Assuming this generic IMBH formation scenario, we calculate the mass functions that correspond to the limiting cases of possible merger trees. Based on estimates for the number density of GCs and taking into account the instrumental sensitivity, we show that current observations do not effectively limit the occupation fraction focc of IMBHs formed by runaway mergers of stellar BHs in GCs. However, we find that if runaway mergers occur steadily throughout the lifetimes of GCs (as opposed to happening mainly early in their lifetimes), then a six-year run of aLIGO at design sensitivity will be able to probe down to focc3% at a 99.9% confidence level, either finding evidence for this formation mechanism, or necessitating others if the fraction of GCs that harbor IMBHs is higher.