Firstly, notice that the condition $\operatorname{det}(A + \operatorname{det}(A) \operatorname{adj}(A)) = 0$ is invariant under conjugation $A \mapsto P A P^{-1}$, and so it suffices to check one element for each similarity class. Each $2 \times 2$ real matrix is similar to either an upper triangular matrix, or a "complex number" matrix.
Upper triangular case
$$A = \begin{pmatrix} \lambda_1 & t \\ 0 & \lambda_2 \end{pmatrix} \in \operatorname{Mat}_{2 \times 2}(\mathbb{R})$$
The condition $\operatorname{det}(A) \neq 0$ gives that $\lambda_1$ and $\lambda_2$ are nonzero. We compute
$$A + (\operatorname{det} A)(\operatorname{adj} A) = \begin{pmatrix} \lambda_1 + \lambda_1 \lambda_2^2 & -t\lambda_1 \lambda_2 \\ 0 & \lambda_2 + \lambda_1^2 \lambda_2 \end{pmatrix}$$
which has determinant $\lambda_1 \lambda_2 (1 + \lambda_1^2) (1 + \lambda_2^2)$. This determinant being zero forces one of $\lambda_1, \lambda_2$ to be zero, which is not allowed. So there are no upper-triangular real matrices $A$ satisfying the conditions.
Complex number case
$$ A = \begin{pmatrix} a & -b \\ b & a \end{pmatrix} \in \operatorname{Mat}_{2 \times 2}(\mathbb{R})$$
In this case the matrix $A$ works like the complex number $z = a + ib$, and we have $\operatorname{det}(A) = z \overline{z}$ and $\operatorname{adj}(A) = \overline{z}$, where $\overline{z} = a - ib$ is the complex conjugate. So
$$\operatorname{det}(A + (\operatorname{det} A)(\operatorname{adj} A)) = |z + z \overline{zz}|^2 = |z|^2 |1 + \overline{z}^2|^2 = 0$$
gives that $z$ must be a square root of $-1$, i.e. one of the two matrices
$$ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$
Conclusion
So the equation only has solutions when $A$ is an invertible $2 \times 2$ matrix with eigenvalues $\pm i$. In this case, it is easy to check (using either of the two matrices above) that $\operatorname{det}(A - (\operatorname{det} A)(\operatorname{adj} A)) = 4$.
I'm not going to do any of the cases with more dimensions, but perhaps this gives you a way to think about what might happen. For example, every $3 \times 3$ real matrix is similar either to an upper triangular matrix, or a "block upper-triangular" matrix where along the diagonal we have a real entry, and then a $2 \times 2$ "complex number" entry.