One such example is $\langle a,b,c \mid a^{-1}ba=b^2, b^{-1}cb=c^2, c^{-1}ac=a^2 \rangle$.
You can use this to construct a sequence of examples of increasing complexity. The example above is the first in the sequence and has total relator length $15$. The second group in the sequence is
$$\langle a,b,c \mid A^{-1}BA=B^2, B^{-1}CB=C^2, C^{-1}AC=A^2 \rangle,$$ where $A=a^{-1}bab^{-2}$, $B=b^{-1}cbc^{-2}$, $C=c^{-1}aca^{-2}$, so the total relator length is $75$. You can then repeat this idea to get further more complicated examples.
If you believe that the first presentation defines the trivial group, then it is not hard to prove that the second one does too. The first group is easily proved trivial by coset enumeration programs, but the second one is much harder.