We present the results of 14 nights of I-band photometric monitoring of the nearby brown dwarf binary, epsilon Indi Ba and Bb. Observations were acquired over 2 months, with a total of close to 42 h of coverage at a typically high cadence of 1.4 min. At a separation of just 0.7 arcsec, we do not resolve the individual components, and so effectively treat the binary as if it were a single object. However, epsilon Indi Ba (spectral type T1) is the brightest known T-type brown dwarf, and is expected to dominate the photometric signal. We typically find no strong variability associated with the target during each individual night of observing, but see significant changes in mean brightness - by as much as 0.10 mag - over the 2 months of the campaign. This strong variation is apparent on a time-scale of at least 2 d. We detect no clear periodic signature, which suggests that we may be observing the T1 brown dwarf almost pole-on, and the days-long variability in mean brightness is caused by changes in the large-scale structure of the cloud coverage. Dynamic clouds will very likely produce lightning, and complementary high-cadence V-band and H alpha images were acquired to search for the emission signatures associated with stochastic 'strikes'. We report no positive detections for the target in either of these passbands.