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I'm looking for an example (if it exists) of two non isomorphic groups with the same profinite completions.

Every reference is well accepted. Thank you.

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    Googling `"same profinite completion"` appears to lead to references that answer the question, e.g., http://mathoverflow.net/a/82238/, https://arxiv.org/pdf/1108.5130.pdf, and http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/04Boileau-Friedl_ver3.pdf are the first 3 hits I got. (I say "appears" and comment because I know nothing about this.) – 2017-02-02

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For a very simple example, consider $\mathbb{Q}$ and the trivial group. There are no nontrivial homomorphisms from $\mathbb{Q}$ to any finite group, so the profinite completion of $\mathbb{Q}$ is trivial.

For a less trivial example, take $\mathbb{Z}$ and its profinite completion $\hat{\mathbb{Z}}$. The group $\hat{\mathbb{Z}}$ is its own profinite completion (since any homomorphism to a finite group must factor through the quotient $\hat{\mathbb{Z}}/n\hat{\mathbb{Z}}\cong \mathbb{Z}/n\mathbb{Z}$ for some $n$), so $\mathbb{Z}$ and $\hat{\mathbb{Z}}$ have the same profinite completion, but they are not isomorphic.