Anywhere with tenement flats is likely to be well into the hundreds if you take a small enough sample area.
Looking at the analysis of census data from the council (below) Leith Walk area peaks out at 52 per acre.
http://www.edinburgh.gov.uk/download/downloads/id/2942/census_2011_-_population_distribution_and_density.pdf
In common with many other cities, population densities in Edinburgh are highest in inner suburban areas surrounding the commercial core of the city centre, notably in the Leith Walk, Fountainbridge and South Side areas. The location with the highest population density in Edinburgh is the Leith Walk area, with a peak of nearly 26,000 people resident within an 800 metre radius (equivalent to a density of 12,900 persons per sq.km.). This is a higher local population density than anywhere else in Scotland, including Glasgow.
Indeed further investigation reveals that Edinburgh contains some of the highest population
densities in the UK outside London, exceeded only by Brighton and Portsmouth..."