The most expensive cities in the world all have cheap neighborhoods. The catch is that the cheap neighborhoods are never the ones the guidebooks recommend. Guidebooks optimize for convenience and safety, which means they send you to the most central, most touristy, most expensive areas. The neighborhoods where locals actually live on normal salaries are one or two metro stops further out, and they are almost always more interesting.
Here is my cheat sheet, developed over fifteen years of traveling on a writer's budget.
London: Peckham and Deptford. Central London hotel prices are obscene — budget travelers have no business staying in Mayfair or Kensington. Peckham, in southeast London, has the best independent food scene in the city (Peckham Levels, Rye Lane market), excellent pubs, and is fifteen minutes from central London by Overground. Deptford, adjacent, is even cheaper and has a thriving weekend market.
Paris: Belleville and the 20th. Le Marais and Saint-Germain are beautiful and unaffordable. Belleville, in the 20th arrondissement, is Paris's most multicultural neighborhood — Chinese, North African, and sub-Saharan African communities side by side — with restaurants serving food from all of those traditions at a fraction of Left Bank prices. The views from the Parc de Belleville are better than from the Eiffel Tower.
New York: Astoria and Jackson Heights, Queens. Manhattan is a financial impossibility for budget travelers. Brooklyn is now nearly as expensive. Queens remains the real New York — and Astoria (Greek tavernas, Egyptian bakeries, the Museum of the Moving Image) and Jackson Heights (the most ethnically diverse square mile in the world) deliver a depth of experience that Manhattan lost decades ago.
Tokyo: Shimokitazawa and Koenji. Shinjuku and Shibuya are where the guidebooks send you. Shimokitazawa (vintage shops, tiny live-music venues, independent cafés) and Koenji (the counterculture capital of Tokyo, with the best used-record stores in Japan) are two stops further on the Chūō line and feel like different cities — slower, cheaper, more human-scaled.
Sydney: Newtown and Marrickville. The harbour neighborhoods are impossibly expensive. Newtown's King Street is Sydney's most interesting eating street — Thai, Ethiopian, Japanese, and Australian-fusion restaurants in a stretch you can walk in twenty minutes. Marrickville, the next suburb south, has Sydney's best Vietnamese food and a growing natural-wine scene.
Copenhagen: Nørrebro. Copenhagen is expensive by any standard, but Nørrebro is the neighborhood where young Danes on normal salaries actually live. Jægersborggade is one of the most interesting food streets in Europe — all independent, all affordable by Copenhagen standards. The Nørrebro Bryghus brewery is worth an evening.
Amsterdam: Amsterdam Noord. The free ferry across the IJ river takes five minutes and lands you in a different world. Amsterdam Noord's NDSM wharf — a former shipyard turned creative district — has street food, studios, and a weekend flea market. Eye Filmmuseum is here. Hotel prices are half of what you would pay in the canal ring.
The overarching principle: in every expensive city, there is a 'value frontier' — the line where touristy-expensive tips over into locally-affordable. That line is usually one or two transit stops from the center, and crossing it improves the trip while halving the cost. Find the frontier. Cross it.