The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are finitely presented.
Are there simpler examples of intermediate growth if we drop the requirement that there exists an inverse?
Yes. Jan Okninski showed that $$\begin{bmatrix} 1 & 1 \\ 0 &1\end{bmatrix}\ \text{and}\ \begin{bmatrix} 1 & 0\\ 1 & 0\end{bmatrix}$$ generate a semigroup of intermediate growth. Details can be found in Nathanson. The growth was estimated there to be like the Hardy-Ramanujan estimate of the partition function. An exact asymptotic growth rate of $e^{\sqrt{n/\log n}}$ was obtained for this semigroup by Lavrik-Mannnlin.
There are also finitely presented examples. The simplest example I know is Yuji Kobayashi. A finitely presented monoid which has solvable word problem but has no regular complete presentation. Theoret. Comput. Sci., 146(1-2):321–329, 1995.. This example is almost cancellative. It has a zero but you can cancel whenever neither product is zero. The growth is essentially the same as Okninski’s example. The first finitely presented example is due to James Shearer (James B. Shearer. A graded algebra with a nonrational Hilbert series. J. Algebra, 62(1):228– 231, 1980) and in a note added in proof he gives essentially the same presentation as Kobayashi but without proof. The generators are $a,b,c,0$ and the relations are $0x=0=x0$ for $x$ any generator and $ab=ba$, $bc=aca$ and $acc=0$.
One can also build monomial examples easily that are not finitely presented. Take any infinite word $w$ over a finite alphabet $A$ whose factor complexity has intermediate growth (these exist for example here) and take the quotient of the free semigroup on $A$ by the ideal of all words not appearing as a factor in $w$. This is I believe the oldest construction due to Govorov I believe.
