During a talk I mentioned in passing Borel's result that for $G$ a connected Lie group, $H^*(BG;\mathbb Q)$ is a polynomial ring. An audience member corrected me in very short order that no, it's in fact a power series ring. I responded that while that statement would follow from a common alternate definition of cohomology, for our limited computational purposes in the rest of the talk the standard Hatcher definition
$$H^* X := \bigoplus H^n X$$
would be unproblematic.
The response was that no, that was still wrong. As this question dragged on, and I ceased being able to imagine the other audience members were still listening, I panicked a bit and came to the somewhat credibility-damaging compromise that my interlocutor could mentally interpose a second pair of brackets whenever we faced a polynomial ring in the sequel. (He was quieted but unappeased.)
1. What does the definition $H^*X := \prod H^n X$ gain us? The examples I know are being able to define the Chern character $K^* \to H^*(-;\mathbb Q)$ and in general the first Chern class for complex-oriented cohomology theories, but I'm hoping for some overarching, systematic, moral reason.
2. What are some real errors produced by $H^* X := \bigoplus H^n X$, such that this gentleman would regard it not as a sometimes-less-convenient convention but a blatant falsehood?
3. Is there any advantage, on the other hand, to retaining the direct sum definition?
4. One encounters as well the convention that a graded ring $A$ is a sequence $(A_n)_{n \in \mathbb Z}$ of abelian groups and bilinear maps $A_m \times A_n \to A_{m+n}$ meeting a list of conditions. Is there a strong reason to prefer this over the direct sum definition, other than to eliminate the question "What is the degree of $0$?"?
This is not a full answer, but maybe gives a hint to the answer of question 1.
An important characteristic class is the Chern character. For (complex) line bundles it is defined as the the formal exponential of the first chern class, i.e. $ch(L)=e^{c_1(L)}:=1+c_1(L)+\frac{c_1(L)^2}{2!}+\ldots$. For higher dimensional vector bundles one defines the chern character by a formal splitting in chern roots. The chern character relates $K$-theory and cohomology, as the exponential has the wonderful property of turning sums into products.
If a cell complex is not finite, the sum, hence the chern character, naturally lives in $\prod_n H^{n}(X;\mathbb{Q})$ and not $\oplus_n H^{n}(X;\mathbb{Q})$. This occurs for the most important chern character of all, the chern character of the tautological bundle over $\mathbb{CP}^\infty$. Note that $\mathbb{CP}^\infty$ is the classifying space of $U(1)$.