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I'm struggle to find a definition of the sheafification which avoids étale spaces, but also Grothendieck topologies.

Note that a preasheaf (of abelian groups) $F$ on a topological space $X$ is a sheaf if and only if it respects, for each open $U\subset X$ and each cover $C$ closed with respect to finite intersections, $F(U)=\underset{V\in C}{\lim}F(V)$ $\{*\}$.

So, can we define the sheafification $F^+$ of $F$ as the sheaf $F^+(U)=\underset{V\subseteq U}{\lim}F(V)$?

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    $U$ is a terminal object of the category of open subsets of $U$, so for any contravariant functor $F$, you have $$\lim_{V \subseteq U} F(V) = F(U)$$ – 2017-02-03
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    You are right... Then, the cover $\{V\}_{V\in U}$ is the trivial one for which the property $(*)$ is satisfied, I'd say that we should rather look for a "minimal" cover, then, which make me think to the following definition: $F(U)=\underset{C\in\Phi}{\operatorname{colim}}\underset{V\in C}\lim F(V)$ where $\Phi$ is the family of all the "good" covers. Too bad this resemble way too much the site approach. – 2017-02-03
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    That formula looks right (which I assume you mean as a definition of $F^+(U)$, not a definition of $F(U)$); it says that an element of $F^+(U)$ is given by patching together a local choice of sections of the presheaf, and that refinements of the patching give the same element. – 2017-02-03
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    It's really not clear to me exactly what you're looking for or trying to avoid. e.g. I could not have predicted you would reject that formula. – 2017-02-03
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    I guess I can't reject that formula! At first sight i just didn't like the fact that there was two limits becouse I hoped for something even easier xD I wonder if we can restrict the colimit over the poset of _finite_ covers ordered by refinement... (if the space $X$ is good enough) – 2017-02-05
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    I think compactness is necessary for that to work. As a counterexample, consider the presheaf $F$ on $\mathbb{R}$ where $F(U)$ is a one-element set whenever $U$ is contained in an interval of length 1, and $F(U)$ is empty otherwise. $F^+$ should be the terminal sheaf, but the global element can't be given by a finite cover. – 2017-02-05

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You can prove the sheafification must exist for more general reasons, if you like. By Theorem 2.48 of Adamek-Rosicky, Locally Presentable and Accessible Categories, any full subcategory of a "locally presentable category," for instance, the category of presheaves on a space $X$, closed under limits and sufficiently filtered colimits is reflective. You can prove very straightforwardly that a limit of sheaves is a sheaf, and as for $\lambda$-filtered colimits, you can take $\lambda$ bigger than the powerset of the topology on $X$, so that $\lambda$-filtered colimits commute with the products and equalizers appearing in the sheaf condition, which implies that a $\lambda$-filtered colimits of sheaves is a sheaf.

There's also an explicit construction very similar to that you might be running into around Grothendieck toposes: the category of sheaves on $X$ is the category of limit-preserving functors from the opposite of the category $\mathcal C$ of open subsets of $X$ to the category of sets. Since we can check whether a presheaf is a sheaf by looking at a small set of limit cones, the construction of 1.36 and the argument of Example 1.33 (8) in the same book give a reflection of preserves into sheaves.

So, using either proof, the existence of sheafification is really a consequence of local presentability of the category of presheaves, and doesn't depend on anything more specific about sheaf/topos theory.