Reduced BF-monoids fit the bill here. 

If $H$ is a multiplicatively written monoid with identity $1_H$, then an *atom* of $H$ is an element $a \in H \setminus H^\times$ for which there do not exist $x, y \in H \setminus H^\times$ such that $a = xy$, where $H^\times$ is the group of units of $H$. In particular, $H$ is called *atomic* if every non-unit of $H$ is a (finite) product of atoms, and is a *BF-monoid* if the factorizations (into atoms) of a fixed element cannot get arbitrarily long. On the other hand, we say that $H$ is *reduced* if $H^\times = \{1_H\}$.

There is a vast literature on the factorization theory of atomic monoids, though most of it is centered on the *commutative* and *cancellative* setting, for which you may want to have a look to:

> A. Geroldinger and F. Halter-Koch, *Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory*, Pure Appl. Math. **278**, Chapman & Hall/CRC, Boca Raton (FL), 2006.

If, on the other hand, you are also interested in non-cancellative or non-commutative monoids, then it's a totally different story, and the best I can do is to address you to my own work with Yushuang Fan, where you will find some pointers to relevant literature (most notably, Smertnig's and Baeth and Smertnig's work on cancellative categories) and an entire section devoted to fundamental aspects of the theory (namely, Sect. 2):

> Y. Fan and S. Tringali, *Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics*, preprint ([arXiv:1701.09152][1]).

Lastly, if you find yourself wondering about sufficient criteria for a monoid to be BF, then there might be just the thing for you in [another thread][2]: Among many others, free monoids and free abelian monoids are BF-monoids (this is obvious), and so is the multiplicative monoid of non-zero elements of a Noetherian integral domain (in particular, the ring of integers of a number field) (this is less obvious).

Of course, reduced BF-monoids are *not* the only semigroups that fit your requests. But, to the best of my knowledge, they are the only class for which a systematic theory of factorization has been so far developed.


  [1]: https://arxiv.org/abs/1701.09152
  [2]: https://mathoverflow.net/questions/264379/