Argthtjtr

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.

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.

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.

$27$ 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.

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:$$2^{30}xge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}cdot 8xge y$$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.

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.