Say a monoid $M$ has infinite products if, for any (possibly infinite) sequence $(m_1,m_2,\ldots)$ of elements of $M$, there exists an element $m_1m_2\cdots\in M$, satisfying some good properties. First, if the sequence is finite, it should coincide with the usual product on $M$. Second, concatenation of sequences results in multiplication of their products and is associative. Third, identities can be "thrown out," as can consecutive inverses ($m_{i+1}=m_i^{-1}$). (Other good properties to include?)

Another way to phrase this is: "$M$ is closed under small ordinal colimits." That is, if $M$ is considered as a one-object category, then for any small ordinal $[\kappa]$ and functor $m\colon[\kappa]\to M$, the colimit of $m$ exists in $M$.

Example: let ${\mathbb N}^+$ denote the monoid with underlying set ${\mathbb N}\cup ${$\infty$} and whose operation on a sequence $m=(m_1,m_2,\ldots)$ is given by addition if $m$ has only finitely many non-zero elements, and by $\infty$ otherwise.

Now suppose that $M$ is any monoid and I want to replace it by a monoid that has infinite products. I'm hoping there are two ways to do this. One would be to add colimits freely, and the other would be to add a single "$\infty$" element that served as a catch-all (as in the example above).

Q: Do these both exist (functorially in $M$)? If so, can you describe them in elementary terms? For example, I'm worried about sequences like $1-1+1-1+\cdots$. So in a good answer I'd hope to see what happens with such infinite sums.