Most of London is quite well served with orbital routes. In my mind the biggest gap is in the Northern quadrant, between the Metropolitan line and the eastern end of the Piccadilly line. There are basically no orbital connections between the ~ 8 lines that head north out of the city, and the overground NLL is very close in to zone 1 at this point too, and most of its connections are out of station interchanges.
A route that started at Barking Riverside, using the eastern end of the GOBLIN, then onto the former
Edgware, Highgate and London Railway (EHL) from around Harringay to Highgate, run alongside the Northern line to Finchley Central - it looks like there is a fair bit of space (probably not enough, but this is speculative forums after all), take over the Mill Hill East branch, and then extend over the old Northern Heights route of the EHL to Mill Hill Broadway and then Edgeware. It would then run, presumably in tunnel via Canons Park and Harrow & Wealdstone to Harrow on the Hill (about a 4 mile tunnel), then join the Uxbridge branch of the Met through West Harrow, before heading down a new curve to the Piccadilly Line Rayners lane branch, heading south through South Harrow. Where it goes from there - there are a few options, ideal would be through Acton Town and on another new curve through Gunnersbury to Richmond.
Although this runs over a lot of busy existing lines, I believe that all of them have some capacity, as they are all outer branches where other trains have already branched off elsewhere, so something like 4tph is probably workable.
It would obviously be massively expensive, but a lot cheaper than boring a tunnel right around London, and would connect pretty much every line in North London, running to the south side of the river at each end (assuming the extension to Abbey Road happens). This would give a lot of useful connectivity, and given that it runs fairly parallel to the North Circular, the demand would probably be good. Obviously it would need to use dual voltage EMUs capable of running on OHL or 3rd / 4th rail. I think they could be S stock sized trains, as the Rayners Lane branch of the Piccadilly Line can just take a subsurface train, so upgrades to make this possible in service are probably practical.
Unrealistic, yes, but probably less so than most ideas on here. And I did enjoy this bit of crayoning!