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My favorite is Thompson's group $V$.

My favorite picture of $V$ is to take a set $X$ which is a disjoint union of subsets $L$ and $R$ having fixed bijections $l:X\rightarrow L$ and $r:X\rightarrow R$. Finite words in $l$ and $r$ map X $X$ to "fragments" of $X$. Two "fragments" $W$ and $U$ that are the images, respectively, of words $w$ and $u$ in $l$ and $r$ are connected by the bijection $uw^{-1}:W\rightarrow U$. The two "fragments" will be disjoint iff neither of $w$ nor $u$ is a prefix of the other. If this condition holds, let $(w,u)$ represent the permutation on $X$ that is $uw^{-1}$ on $W$, is $wu^{-1}$ on $U$, and is the identity elsewhere. The group $V$ is generated by all such $(w,u)$. It is finitely presented and contains all finite groups.

Other f.p. groups containing all finite groups are known as Houghton groups. See the section on Houghton groups K. S. Brown "Finiteness properties of groups" in Journal of Pure and Applied Algebra, 44 (1987), 45-75. I forget how they are indexed, but from about n=3 on up, they are all f.p.

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My favorite is Thompson's group $V$.

My favorite picture of $V$ is to take a set $X$ which is a disjoint union of subsets $L$ and $R$ having fixed bijections $l:X\rightarrow L$ and $r:X\rightarrow R$. Finite words in $l$ and $r$ map X to "fragments" of $X$. Two "fragments" $W$ and $U$ that are the images, respectively, of words $w$ and $u$ in $l$ and $r$ are connected by the bijection $uw^{-1}:W\rightarrow U$. The two "fragments" will be disjoint iff neither of $w$ nor $u$ is a prefix of the other. If this condition holds, let $(w,u)$ represent the permutation on $X$ that is $uw^{-1}$ on $W$, is $wu^{-1}$ on $U$, and is the identity elsewhere. The group $V$ is generated by all such $(w,u)$. It is finitely presented and contains all finite groups.

Other f.p. groups containing all finite groups are known as Houghton groups. See the section on Houghton groups K. S. Brown "Finiteness properties of groups" in Journal of Pure and Applied Algebra, 44 (1987), 45-75. I forget how they are indexed, but from about n=3 on up, they are all f.p.