To find how many three-digit numbers can be expressed as an integral power of $2$, we need to evaluate numbers of the form $2^n$ (where $n$ is an integer) and check which ones fall in the range of $100$ to $999$.

Let us list the positive integral powers of $2$:

• $2^1 = 2$
• $2^2 = 4$
• $2^3 = 8$
• $2^4 = 16$
• $2^5 = 32$
• $2^6 = 64$
• $2^7 = 128$ (Three-digit number)
• $2^8 = 256$ (Three-digit number)
• $2^9 = 512$ (Three-digit number)
• $2^{10} = 1024$ (Four-digit number)

As we can see, the powers of $2$ continue to grow, and any power greater than $9$ will result in a number with four or more digits. Negative integral powers of $2$ result in fractions (e.g., $2^{-1} = 0.5$), which are not three-digit numbers.

Thus, the only three-digit numbers that are integral powers of $2$ are $128$, $256$, and $512$.

There are exactly $3$ such numbers.