A hobby-horse is as mechanically simple as it gets. Two wheels and a seat, even steering is optional. Many of them did not even have padded wheels, "solid" meant iron. Today these are built for children who are learning to ride (the "like-a-bike" is one such product).
Next is an ordinary (or penny-farthing) There's only five axles and the whole thing can be made using basic tools. These are still made today but primarily as replicas of historical bikes. There are national and world championship competitions. The modern "tall bike" is in many ways similar but often has gears, brake and so on so lacks the simplicity of the ordinary bike.
For a modern bicycle, a fixed gear bike is as simple as you're likely to want to ride. Chain drive, but no freewheel and brakes are not strictly necessary (but may be required by law). Generally they have ball bearings rather than plain bearings, but they're quite simple to assemble or fix compared to other modern bikes. However, since a freewheel is a single part as far as most bike machanics are concerned a BMX or trials bike is effectively as simple to deal with. Especially "street BMX" bikes that often have back-pedal hub brakes. The 406 wheels and lack of rim brakes mean that the wheels are very strong and don't have to be especially true. Most fixed gear bikes have limited clearance around the wheels so they have to be quite close to true or they hit the frame.
Once you add gears, hub gears are simple for the owner both in operation and maintenance. With wide, straight chain, full chain case and commonly also hub brakes mean there's little maintenance to do. Hub gears have a single shifter and the gears are "in order", meaning that shifting up always produces a higher gear, and it's always the next higher gear. This makes the bike very simple to operate. With a derailleur system there's usually two gear levers (for the front and rear derailleurs), and the shift pattern to go from lowest to highest gear is non-trivial (and varies between bikes).
Likewise, hub brakes usually require less maintenance and are more effective than rim brakes. They're also heavier and more expensive rhan rim brakes, but if your criteria is simple to operate, hub brakes win. Maintenance is generally infrequent, with cable operated disks requiring most frequent maintenance and some hub and hydraulic disk brakes being almost maintenance free (lasting decades of use in gentle city riding).
It's not unheard of to fish a 50 year old bike out of a garage (or canal) and find that the hub brakes and gears still work.