Chain line issue

When you say ‘top three gears’ do you mean the biggest cogs on the cassette, or the top three gears (as in highest ratios - the smallest cogs on the cassette)?

So, is the chain coming off the chainring at the front or off the sprockets at the back?

First thing I would do is adjust the rear derailleur, especially the limit screw adjustments to ensure it is shifting correctly.

Check the chain for wear, easiest method is with a chain checker, but you can measure it with a steel rule. If it is 3/4 worn or more then it will likely be jumping and / or starting to damage your sprocket/ chainring teeth.

Now check the chainrings and cassette teeth for excessive wear and/ or damage. Search “Sheldon Brown’s” website which has descriptions of tooth wear.

Now, I am assuming that amongst that list you will probably have found the cause. If not, other things to consider are;

Derailleur hanger for straightness/ alignment
Chainring bolts for tightness
Chain line (correct bottom bracket fitment)
Rear derailleur cage for twist
Rear derailleur pulley wheels for wear (teeth worn excessively, bearings/ bushings worn/ right) - bear in mind the top pulley will have some play
Shifter cable for smoothness of operation - check for kinks, fraying, corrosion or stickiness in operation
Check shifter for correct function

And that’s about all I can think of off the top of my head...

Pictures would help...

PP

I once set up a frankenbike 1x8 from an old frame. However, the frame was an old steel frame that needed the rear dropouts cold-set wider to allow for the wider wheel - and this meant the sprockets were further outboard than they should have been, so going to the smaller sprockets would have the chainring drop the chain.

Yours is doing the opposite, so it sounds like your chainset is too far outboard. If it's an older square taper bottom bracket? There are so many different versions (JIS and ISO standard tapers, widths for track, single, double and triple chainrings) that the wrong one will put the chainline out of whack.

Finally, is the mech hanger or the rear mech pulley cage bent? If it's really out, back-pedalling may ship the chain from the chainring too.