Is there an example of a finitely generated (infinite) residually finite group $\Gamma$ for which every linear representation of $\Gamma$ has finite image?
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A group $G$ is just infinite if it is infinite but every proper quotient is finite. Clearly a just infinite group which is not linear has the property that its image under any linear representation is finite. Thus any group which is finitely generated, residually finite, not linear and just infinite is an example of what you want: for instance, the Grigorchuk group. 

