The Eilenberg-Mazur swindle shows that the Grothendieck group of an additive category with countable coproducts is trivial. The strategy is to observe that any "Euler characteristic" $\chi$ on such a category must be zero, because for any object $P$, we have $$P \oplus \bigoplus_{i=1}^\infty P \simeq \bigoplus_{i = 1}^\infty P$$ which implies that $\chi(P) + \chi(Q) = \chi(Q)$ for $Q = \bigoplus_{i=1}^\infty P$.

Is there any analog for the higher $K_i$ (of, say, an exact category in Quillen's sense)? I don't know any simple way of thinking of the higher $K_i$ (e.g. via Euler characteristics), but it would be interesting if, say, the associated K-theory space somehow had to be contractible.