A matrix with a single eigenvalue

Eigenvalue. $XI-A$ is upper triangular with all diagonal entries $X-3$, so $\chi_A(X)=(X-3)^3$: the only eigenvalue is $3$, with algebraic multiplicity $3$.

Eigenspace.
$$A-3I = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.$$
$(A-3I)(x,y,z)^T=(y,z,0)^T=0$ iff $y=z=0$, so $E_3=\{(x,0,0)\}$, of dimension $1$.

Conclusion. The geometric multiplicity $1$ is smaller than the algebraic multiplicity $3$, so $A$ is not diagonalizable. (Directly: a diagonalizable matrix whose only eigenvalue is $3$ equals $3I$.)

Remark. $A$ is a single Jordan block of size $3$ for the eigenvalue $3$.