One lily pad, doubling in size every day, covers a pond in 30 days. How long would it take eight lily pads to cover the pond? 
A lily pad sits on a pond. It doubles in size every day. It takes 30
  days for it to cover the pond. If you start with 8 lily pads instead,
  how many days does it take to cover the pond?

I think that the answer is $27$, but I don't really think that makes sense intuitively. I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.
 A: $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after three days), and how long it takes it to cover the lake from there.
Also, this relative "indifference" to a seemingly large disparity in starting point ("$x$ times more to start with means it takes $30-y$ days rather than $30/y$") is exactly what exponential growth means.
A: Your answer is correct
If we denote the size of the lily pad by $x$, then after $1$ day, the coverage becomes $2x$ , . . . Also let's denote the size of pond by $y$. The assumptions imposes that:$$2^{30}x\ge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}\cdot 8x\ge y$$and$$2^{26}\cdot 8x< y$$which shows that the completion of the process takes long as much as $27$ days. 
The intuition is that:
If we start with $1$ lily pad, after $3$ days we will have $8$ of them since the size of lily pads doubles each day. Starting with $8$ lily pads in first place and observing their growth is equivalent to observe the growth of $1$ lily pad during the days skipping the first $3$ days. If whole the process takes long $30$ days, then or modified process takes long $30-3=27$ days which is totaly intuitive.
A: Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.
A: Answering the 'why is my intuition wrong?' aspect:
The key thing to remember here is not everything is linear. The unrealistic nature of the question isn't what makes it unintuitive. It's that we have a tendency to think things are linear, even when we know full well they are not.
The reasoning used seems to be 'if I start with twice as much, it must only take half as much time to get to the same end amount'. That's using linearity, and basic ideas of addition/multiplication. But exponential growth is very very much not like this.
There are other places this comes up in life: if you are speeding up then you cover most of the ground at the end; if you are saving for your pension, a large part of your savings comes from the last few years (when your salary is highest); twins newly separated are only half the size of a single baby the same age, but they'll only take minutes longer to reach full size, not twice as long.
A: Hint $\#1$:
At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.
In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?
(I know you've already solved this, I just think rewording/reframing the question might make it a bit easier to grasp on the intuitive level.)

Hint $\#2$:
If that doesn't help ease your intuition behind your answer (which to my understanding is correct), keep in mind that starting with $8$ lilypads is basically no different than your first scenario after $3$ days. Sure, you have more lilypads, but since each doubles in size, it's no different than one lilypad of the same size as those $8$ put together then doubling. The number of lilypads differs, but we're focused on the total area encompassed.
A: Another way to look at it is to work backward.
First just consider the one lily pad. After $29$ days it covers half the pond. After $28$ days a quarter of the pond. After $27$ days an eighth of the pond.
So after $27$ days eight lily pads would cover the whole pond.
A: 
I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.

Lily pad doubles in size every day, so it is increasing as a geometric progression. 
$$\begin{array}{c|c|c|c|c|c|c|c|c}
\text{n-th day}&1&2&3&4&\cdots&26&27&28&29&30\\
\hline
\text{size of $1$ lily pad}&1&2&2^2&2^3&\cdots&2^{25}&2^{26}&2^{27}&2^{28}&\color{red}{2^{29}} \end{array}$$
If you start with $8$ lily pads, each doubling on its own, then:
$$\begin{array}{c|c|c|c|c|c|c|c|c}
\text{n-th day}&1&2&3&4&\cdots&26&27&28&29&30\\
\hline
\text{size of $8$ lily pads}&2^3&2^4&2^5&2^6&\cdots&2^{28}&\color{red}{2^{29}}&2^{30}&2^{31}&2^{32} \end{array}$$
Because when each of $8$ lily pads keeps doubling per day, the $8$ lily pads increase $8$ times faster in size altogether than that of one lily pad. So, you must multiply the size of one lily pad on any day by $8=2^3$ to find the total size of $8$ lily pads.
As this source informs, the Giant Water Lily may grow as large as $8$ to $9$ feet ($2.4-2.7$m) in diameter.   
