We need to order the following numbers from least to greatest without using a calculator: $2$, $\sqrt{3}$, and $\sqrt{\pi}$.

1. Problem Description

We need to order the following numbers from least to greatest without using a calculator: $2$, $\sqrt{3}$, and $\sqrt{\pi}$.

2. Solution Steps

First, let's approximate the value of $\pi$. We know that $\pi \approx 3.14$.

Now, we have the numbers $2$, $\sqrt{3}$, and $\sqrt{\pi} \approx \sqrt{3.14}$.

We can square each number to compare them more easily.

$2^2 = 4$

$(\sqrt{3})^2 = 3$

$(\sqrt{\pi})^2 = \pi \approx 3.14$

Since $3 < 3.14 < 4$, we have $\sqrt{3} < \sqrt{\pi} < 2$.

3. Final Answer

The expressions ordered from least to greatest are $\sqrt{3}, \sqrt{\pi}, 2$.