<s>[INST] Unique least square solutions There is a theorem in my book that states: If $A$ is $m\times n$, then the equation $Ax = b$ has a unique least square solution for each $b$ in $\mathbb{R}^m$. 
But can we find a counter-example to this by providing a matrix $A$ and vector $b$ such that $A^TAx = A^Tb$ produces a general solution with a free variable?
 [/INST] Your theorem statement is incomplete. Requirements have been omitted.
To amplify the insights of @Troy Woo, given a matrix $\mathbf{A}\in\mathbb{C}^{m \times n}$, a solution vector $x\in\mathbb{C}^{n}$, and a data vector $b\in\mathbb{C}^{m}$ such that $b\notin\mathcal{N}(\mathbf{A}^{*})$, and where $n\in\mathbb{N}$ and $m\in\mathbb{N}$, the linear system
$$
\mathbf{A} x = b
$$
has the least squares solution can be expressed in terms of the Moore-Penrose pseudoinverse $\mathbf{A}^{\dagger}$:
$$
  x_{LS} = \mathbf{A}^{\dagger}b + \left(\mathbf{I}_{n} - \mathbf{A}^{\dagger}\mathbf{A} \right) y
$$
with the arbitrary vector $y\in\mathbb{C}^{n}$. 
If the matrix rank $\rho < m$, the null space $\mathcal{N}\left(\mathbf{A}\right)$ is non-trivial and the projection operator $\left(\mathbf{I}_{n} - \mathbf{A}^{\dagger}\mathbf{A} \right)$ is non-zero.
Example
The linear system
$$
\begin{align}
  \mathbf{A} x & = b \\
%
  \left[
    \begin{array}{cc}
      1 & 0 
    \end{array}
  \right]
%
  \left[
    \begin{array}{c}
      x_{1} \\
      x_{2}
    \end{array}
  \right]
%
&=
%
  \left[
    \begin{array}{c}
      b_{1} 
    \end{array}
  \right]
\end{align}
$$
has the least squares solution
$$
\begin{align}
   x_{LS} & = \mathbf{A}^{\dagger} b + \left( \mathbf{I}_{2} - \mathbf{A}^{\dagger} \mathbf{A}\right) y\\
%
&=
%
  \left[
    \begin{array}{c}
      b_{1} \\
      0
    \end{array}
  \right]
%
+
%
\alpha 
  \left[
    \begin{array}{c}
      0 \\
      1
    \end{array}
  \right]
\end{align}
$$
with $\alpha \in \mathbb{C}^{n}$.
The affine space of the solution satisfies
$$
\mathbf{A} \left(
  \left[
    \begin{array}{c}
      b_{1} \\
      0
    \end{array}
  \right]
%
+
%
\alpha 
  \left[
    \begin{array}{c}
      0 \\
      1
    \end{array}
  \right]
 \right) = 
%
\mathbf{A} \left(
  \left[
    \begin{array}{c}
      b_{1} \\
      0
    \end{array}
  \right]
 \right)
%
+
%
 \alpha
\mathbf{A}
\left( 
  \left[
    \begin{array}{c}
      0 \\
      1
    \end{array}
  \right]
 \right) = 
%
\mathbf{A} \left(
  \left[
    \begin{array}{c}
      b_{1} \\
      0
    \end{array}
  \right]
\right).
$$
The solution vector of least norm, 
$$\Bigg\lVert 
  \left[
    \begin{array}{c}
      b_{1} \\
      0
    \end{array}
  \right]
%
+
%
 \alpha
  \left[
    \begin{array}{c}
      0 \\
      1
    \end{array}
  \right]
\Bigg\rVert_{2}^{2}$$
corresponds to $\alpha=0$.
</s>