The species given below that does NOT show disproportionation reaction is:

Disproportionation is a redox reaction in which the same element is simultaneously oxidised and reduced, giving two different products with different oxidation states.

For a species to undergo disproportionation, the element must be in an intermediate oxidation state that can go both to a higher and a lower oxidation state.

BrO$$^-$$ has Br in $$+1$$ oxidation state: it can disproportionate as $$3\text{BrO}^- \rightarrow \text{BrO}_3^- + 2\text{Br}^-$$, where Br goes to $$+5$$ and $$-1$$.

BrO$$_2^-$$ has Br in $$+3$$ oxidation state: it can disproportionate as $$2\text{BrO}_2^- \rightarrow \text{BrO}_3^- + \text{BrO}^-$$, with Br going to $$+5$$ and $$+1$$.

BrO$$_3^-$$ has Br in $$+5$$ oxidation state: it can disproportionate as $$4\text{BrO}_3^- \rightarrow 3\text{BrO}_4^- + \text{Br}^-$$, with some Br going up to $$+7$$ and one Br going down to $$-1$$.

BrO$$_4^-$$ has Br in $$+7$$ oxidation state, which is the maximum possible oxidation state for bromine. Since there is no higher oxidation state available for Br, BrO$$_4^-$$ cannot be oxidised further and therefore cannot undergo disproportionation.

The species that does NOT show disproportionation is BrO$$_4^-$$, which is option 1.