Short answer: On a strict $10/day budget, you can survive in 50+ cities. Realistically, $20-$30/day is comfortable for most backpackers.
TenDollarPick uses $10 as a creative constraint. Real-world backpackers usually budget $20-$40/day in Southeast Asia, $40-$80 in Europe, $80-$150 in Japan/Western Europe. The site catalogs which cities are realistic on tight budgets.