I can't speak for others, but the only thing my EU2000i won't run is my 13.5K Btu AC unit, a Dometic model. It will run easy start 11k units, as well as some of the smaller 8k or 9k Coleman polartec models.
My Eu2000i will run my microwave oven.
My Eu1000 won't run much of anything except lights, heater that burns propane, and the battery charger. Not even a hair dryer.
YMMV.
The Eu1000i is most sensitive to crappy gas laden with alcohol clogging it's jets and being a PIA to start and then start hunting or searching with fluctuating rpm's to maintain a load, due to the carb size and main jet size being the smallest. You have to load the gas you buy up with Seafoam and Techron and Blue Marine Sta-bil to keep the jets clean, and turn your gas over frequently, as well as drain the carb anytime you won't be using the /Eu1000i for more than a week.
This can mean more frequent carb tear downs, rebuilds, or flat out replacements. It keeps you broke and small gardener shops very busy sucking money out of your wallet to fix or repair, if you yourself are not mechanically inclined.