Is there some universal sense of -ification (eg, groupification) in category theory I have three questions.
1:
Does the groupification of a semigroup always exist? I believe this should be yes because for every $x$ in the semigroup one could just define an element $x'$ that should work as its inverse. But what would then happen to the product $x'y$ for $x,y$ elements of the semigroup? It feels like we get choices (or maybe not) here that messes things up. 
2:
When defining the groupification, $G$, of a semigroup $S$ one require it to come with a morphism (of semigroups) $S \rightarrow G$ such that any other morphism (of semigroups) from $S$ to another group $G'$ factorizes through the previous map. Exactly which type of objects can be groupified? I guess one cannot groupify a topological space.
3: 
This is a broad question but is there some sense of -ification? In the example one could replace "group" by "topological space" and talk about topologyfication. Now, no such word seem to exist so I guess one could not "topologyfy". 
We can (i think) consider the groupification functor from the category of semigroups to the category of groups and it should be adjoint to the forgetful functor from the category of groups to the category of semigroups. This would suggest that we need some sense of a forgetful functor in the first place to talk about a -ification. 
Apologies for this bad question, sometimes asking the right question is just as hard as answering it. 
 A: I take the liberty of answering only the parts of your question, to which I think I can give a precise answer.
1: Yes, it is called the Grothendieck group $G(N)$ of the given semigroup $S$. This construction is functorial.
2: There is always a canonical semigroup homomorphism $S\to G(S)$, but this need not be injective in general. For example, the Grothendieck group corresponding to $\mathbb N\cup\{\infty\}$ with the obvious addition $(n+\infty=\infty)$ is trivial!
3: Concerning -ification in general, at the moment I have nothing to add to BBischof's great comments above.
A: As BBischof says, the standard notion of -ification is to take the left adjoint of a forgetful functor.  This includes the following as special cases:


*

*The groupification of a monoid or semigroup,

*The free group on a set, the free vector space on a set, etc.

*The abelianization of a group,

*The group ring of a group (the forgetful functor here sends a ring to its group of units),


and many other examples.  I do not think one can reasonably talk about universal -ification without a specific choice of forgetful functor; if no good choice exists, you won't get a good notion of -ification, and on the other hand there may be more than one choice.
(Notably, I remember reading that in at least one example the natural construction is to take the right adjoint, but I don't remember what this example was.)
A: To add to Rasmus's answer to 1. One can't avoid elements like
$x^{-1}y$ in the Grothendieck group. Consider say the natural numbers.
It's Grothendieck group is the integers. As it's additive we
write say four plus the inverse of 6 as $4-6$. Things aren't too
bad in the commutative case: (in additive notation) all elements
of $G(S)$ have the form $a-b$ with $a$, $b\in S$. Moreover
$a-b=c-d$ iff there is $s\in S$ with $a+d+s=b+c+s$
(and we can forget about $s$ if $S$ is cancellative).
But things are nastier in the noncommutative world. Returning
to multiplicative notation, you now get elements in $G(s)$ like $ab^{-1}cd^{-1}e$
which perhaps don't simplify to shorter ones. And it's not so
easy to give a criterion for when two elements are the same
(or equivalently when one element equals the identity).
C'est la vie!
A: The name of the really really general construction (which is indeed in many cases the left adjoint of the forgetful functor) is called the completion of an object. The basic idea is to find the 'most natural' or 'smallest' object having the original as a subobject, which means we can extend the idea to, say, the closure of a subset of a topological space.
Here's the n-lab page if you're interested...
