What ignites on the neutron star of 4U 0614+091?

[abridged] The LMXB 4U 0614+091 is a source of sporadic thermonuclear (type I) X-ray bursts. We find bursts with a wide variety of characteristics in serendipitous wide-field X-ray observations by EURECA/WATCH, RXTE/ASM, BeppoSAX/WFC, HETE-2/FREGATE, INTEGRAL/IBIS/ISGRI, and Swift/BAT, as well as pointed observations by RXTE/PCA and HEXTE. Most of them reach a peak flux of ~15 Crab, but a few only reach a peak flux below a Crab. One of the bursts shows a very strong photospheric radius-expansion phase. This allows us to evaluate the distance to the source: 3.2 kpc. The burst durations are between 10 sec to 5 min. However, after one of the intermediate-duration bursts, a faint tail is seen to at least ~2.4 hours after the start of the burst. One very long burst lasted for several hours. This superburst candidate was followed by a normal type-I burst only 19 days later. This is, to our knowledge, the shortest burst-quench time among the superbursters. A superburst in this system is difficult to reconcile if it accretes at ~1% L_Edd. The intermediate-duration bursts occurred when 4U 0614+091's persistent emission was lowest and calm, and when bursts were infrequent (on average one every month to 3 months). The average burst rate increased significantly after this period. The maximum average burst recurrence rate is once every week to 2 weeks. The burst behaviour may be partly understood if there is at least an appreciable amount of helium present in the accreted material from the donor star. If the system is an ultra-compact X-ray binary with a CO white-dwarf donor, as has been suggested, this is unexpected. If the bursts are powered by helium, we find that the energy production per accumulated mass is about 2.5 times less than expected for pure helium matter.