Limits are additive, but suprema and infima aren't? I find it curious that limits have the additive property, but suprema are only subadditive and infima are only superadditive. Since extrema are limits, why is this so?
I was thinking if, for uniformly bounded functions all defined over the same set of finite measure, the supremum over such functions is additive, not just subadditive. Is this true? Are there more general conditions for functions where suprema and infima are additive over?
 A: You're comparing apples to oranges. Limits are taken over sequences, while supremum and infimum are defined for sets. This is where the concepts of $\lim\sup $ and $\lim\inf$ are most useful. But in general you won't have full additivity for supremum limits and infimum limits in sequences in a similar way to how interference in wave patterns can be either destructive or constructive. If you have the sequence:
$$1,0,1,0,1,0,1,0,...$$
and the sequence
$$0,2,0,2,0,2,0,2,...$$
Then their sum is:
$$1,2,1,2,1,2,1,2,...$$
Because the "lows" of one coincide with the "highs" of the other, and if you notice the lim sup and lim inf are not additive. 
A: There's a sense in which $\limsup$ is a limit. If $\{a_n\}$ is a sequence, then $$\limsup_{n\to\infty} a_n =\lim_{n\to\infty} \left(\sup_{m>n} a_m\right)$$
Now, limits are additive, but that just means that:
$$\limsup_{n\to\infty} a_n + \limsup_{n\to\infty} b_n = \lim_{n\to\infty} \left(\sup_{m>n} a_m + \sup_{m>n} b_m\right)$$
Unfortunately, $$\sup_{m>n} a_m + \sup_{m>n} b_m \neq \sup_{m>n} (a_m+b_m)$$
in general.  We can only say:
$$\sup_{m>n} a_m + \sup_{m>n} b_m \geq \sup_{m>n} (a_m+b_m)$$
A: Maybe the crux here is that additivity of limits assumes existence of limits, which is a stronger condition than existence of extrema or the lim sup/inf. Therefore we can also make a stronger statement.
Both previous arguments are certainly valid, but only to a certain extent:
It is comparing apples to oranges, if you see both notions defined on the biggest possible structure they make sense on, but there are certainly structures that allow the definition of both notions, e.g. sequences or countably infinite sets. 
It is also true that extrema are not limits if you go for the most common definition, but if you allow elements of the set to be chosen multiple times, extrema are limits. 
tldr: A lot of the discussion boils down to what exactly we are actually talking about.
