If your taxes are "near" maximum, then you're running with taxes in the 18-20% region, which will not only dramatically reduce demand for R$-R$$$ but also have a knock on effect in that your current population will also begin leaving, resulting in less available potential workers for both commercial and industrial buildings.
At a guess - since you've stated that you have your taxes at maximum as a method of making money - did you build your entire city infrastructure before allowing people to move in or something? Essentially, if you're at the point where you've built hospitals, schools, police and fire, but are not making a profit, then you're probably trying to expand quicker than your population is increasing, resulting in far more monthly expenditure than your income.
To increase residential demand, ensure that your residential zones are some distance away from buildings which are a source of pollution - power plants, industrial zones, etc - that you have a good road network between your residential, commercial and industrial zones to keep the commute time down, and reduce your taxes for the residential zone type you wish to create demand for. You can raise your taxes right from the start of the game from the starting 9% to around 11-12% but going any higher than this without the luxuries of happy citizens, parks, stadiums, etc will simply result in your population moving out and your zone demands plummeting.
