jmc

Let the variable $r$ represent the number of cherry danishes and $s$ represent the number of cheese danishes. We have $$2s\ge r\ge 3+\frac{2}{3}s.$$The left side of the inequality must be greater than or equal to the right side, so we have $$2s\ge 3+\frac{2}{3}s\Rightarrow 6s\ge 9+2s\Rightarrow 4s\ge 9\Rightarrow s\ge \frac{9}{4}.$$The least possible integer that $s$ can be is 3. The least possible value of $r$ is $$r\ge 3+\frac{2}{3}(3)\Rightarrow r\ge 5.$$Therefore, the student brings at least $\boxed{8}$ danishes.