Jeremy has three cups. Cup $A$ has a cylindrical shape, cup $B$ has a conical shape, and cup $C$ has a hemispherical shape. The rim of the cup at the top is a unit circle for every cup, and each cup has the same volume. If the cups are ordered from least height to greatest height, what is the ordering of the cups?
1. **Determine the volume of each cup:**
   - **Cup A (Cylindrical shape):**
     The volume \( V \) of a cylinder is given by:
     \[
     V = \pi r^2 h
     \]
     Given the radius \( r = 1 \), the volume becomes:
     \[
     V = \pi (1)^2 h = \pi h
     \]

   - **Cup B (Conical shape):**
     The volume \( V \) of a cone is given by:
     \[
     V = \frac{1}{3} \pi r^2 h
     \]
     Given the radius \( r = 1 \), the volume becomes:
     \[
     V = \frac{1}{3} \pi (1)^2 h = \frac{1}{3} \pi h
     \]

   - **Cup C (Hemispherical shape):**
     The volume \( V \) of a hemisphere is given by:
     \[
     V = \frac{1}{2} \left( \frac{4}{3} \pi r^3 \right)
     \]
     Given the radius \( r = 1 \), the volume becomes:
     \[
     V = \frac{1}{2} \left( \frac{4}{3} \pi (1)^3 \right) = \frac{2}{3} \pi
     \]

2. **Equate the volumes to find the heights:**
   Since all cups have the same volume, let the common volume be \( V \).

   - For Cup A (Cylinder):
     \[
     \pi h = V
     \]
     Solving for \( h \):
     \[
     h = \frac{V}{\pi}
     \]

   - For Cup B (Cone):
     \[
     \frac{1}{3} \pi h = V
     \]
     Solving for \( h \):
     \[
     h = \frac{3V}{\pi}
     \]

   - For Cup C (Hemisphere):
     \[
     \frac{2}{3} \pi = V
     \]
     Solving for \( V \):
     \[
     V = \frac{2}{3} \pi
     \]
     Since \( V \) is the same for all cups, we can use this value to find the heights of Cups A and B.

3. **Calculate the heights using the common volume \( V = \frac{2}{3} \pi \):**
   - For Cup A (Cylinder):
     \[
     h = \frac{V}{\pi} = \frac{\frac{2}{3} \pi}{\pi} = \frac{2}{3}
     \]

   - For Cup B (Cone):
     \[
     h = \frac{3V}{\pi} = \frac{3 \cdot \frac{2}{3} \pi}{\pi} = 2
     \]

   - For Cup C (Hemisphere):
     The height of a hemisphere is simply the radius, which is \( 1 \).

4. **Order the cups by height:**
   - Cup A (Cylinder): \( \frac{2}{3} \)
   - Cup C (Hemisphere): \( 1 \)
   - Cup B (Cone): \( 2 \)

Thus, the order from least height to greatest height is:
\[
\boxed{\text{A, C, B}}
\]