Intuitively, why is compounding percentages not expressed as adding percentages? I pursue only intuition; please do not answer with formal proofs. I already know the theoretical reason: because  each percentage expresses a different base. $1.$ But why not intuitively? 
My problem: Whenever adding percentages, I am always initially tempted to add them as cardinal numbers, before resisting myself and spending $\geq 5$  minutes recollecting the following algebra and surmounting the temptation, all of which reveal chasms in  my comprehension. 
$\bbox[5px,border:2px solid gray]{ \text{ Optional Reading and Supplement: } }$
If the price of apricots ($a$) increases by $p_a$  and the price of cherries ($c$) increases by $p_c$, where $0 \le p_a,p_c \leq 1$; then the $\color{green}{ \text {Correct new price =}    (1 + p_a)a + (1 + p_c)c. \tag{2}}$
But adding the percentages as cardinal numbers produces the: $\color{darkred}{ \text { Incorrect new price =}    (1 + p_a + p_c)( a + c) = \color{green}{(1 + p_a)a + (1 + p_c)c} \color{#FF4F00}{ + p_ac + p_ca.} \tag{3}}$
The existence of the 2 orange terms proves $2 \neq 3$, but do not reveal the intuition.
PS: This question is motivated by the first sentence of this quote in this question.
 A: Say something costs $\$100$ and the price goes up by $50\%$.
$50\%$ of $\$100$ is $\$50$, so the new price is $\$150$.
Then it goes up by $50\%$ again.
$50\%$ of $\$150$ is $\$75$, so the new price is $\$225$.
The point is that the second time, you're taking $50\%$ of a larger quantity.
A: 
So, the classic case: You have a 20% off coupon, and the item is
  marked off 25% from regular price. What’s your actual savings at the
  register? Most people immediately add 20% and 25% and arrive at 45%
  off, immediately running to the register.

The absolute saving is $p_1 P_1+p_2 P_2$, the new percentage is
$$
p = \frac{p_1 P_1 + p_2 P_2}{P_1 + P_2}
$$
if the prices were the same $P_1=P_2=P$ the absolute saving would be $(p_1 +p_2)P$ and the percentage
$$
p = \frac{p_1 P + p_2 P}{P + P} = \frac{p_1+p_2}{2}
$$ 
which is the average of the two old percentages.
For other problems we might end up with different relationships between absolute savings and saving in percent. The problem by Michael Hardy, which is iterated prices increases is such a problem.
If you add $20\%$ and $30\%$ of a pizza slice, you talk about $50\%$ of the pizza, so in this context they add.
In summary: You can not expect an additive behaviour of percentages over many different problems, just because some involved quantities are expressed as percentages.
