On what group are you studying the monodromy? Begin with $\mathcal{Y} = \Delta \times \mathbb{P}^n$, $n\geq 2$, with its projection $\text{pr}_1$ to $\Delta$. The central fiber is smooth of multiplicity $1$. Let $d\geq 2$ be an integer. Let $Z\subset \mathcal{Y}$ be a closed submanifold such that the restriction $\text{pr}_1|_Z:Z\to \Delta$ is a $d$-to-$1$ branched cover of the disk by the disk totally ramified over the origin with ramification profile $(d_1,\dots,d_m)$, i.e., $m$ preimages of the origin with the pullback of a coordinate vanishing to order $d_r$ at the $r^\text{th}$ preimage point ($d_1+\dots +d_m = d$). Let $\mathcal{X}'\to \mathcal{Y}$ be the blowing up along $Z$. There is a further blowing up $\mathcal{X}\to \mathcal{X}'$ with center in the special fiber so that $\mathcal{X}$ is smooth with normal crossings special fiber.

The strict transform of the special fiber of $\mathcal{Y}$ is an irreducible component with multiplicity $1$ of the special fiber of $\mathcal{X}$. Thus the GCD of the multiplicities equals $1$. Yet the monodromy action of $\pi_1(\Delta^*,\text{point})$ on $H^{2r}(X)$ is non-unipotent for $1\leq r \leq n-1$. On the other hand, the LCM of the multiplicities is divisible by the LCM $\ell$ of $(d_1,\dots,d_m)$, and the $\ell^\text{th}$ power of a generator of $\pi_1(\Delta^*,\text{point})$ does act unipotently (in fact it acts as the identity, in this case).