Addendum added that responds to the OP's comment question.
I agree with quasi's answer, but wish to express it differently.
Your equation of
$$[(1+3)^{1/9}-1]=16.65\%$$
is wrong. The fund quadruples every three years. Therefore, after 9 years, the fund has increased from $(1)$ to $(64)$, which is an increase of $(63)$.
Therefore, your equation should be
$$[(1+63)^{1/9}-1]=58.74\%$$
Addendum
Responding to the OP's comment question.
Why is it $63$? $4^3$ would be the total return,
but what about the three payments of 1?
After receiving your comment/question, I re-read your original posting
and discovered that I apparently mis-interpreted your intent. As I now
read your original question, you intend the following:
Invest $(10,000)$ into a fund.
After 3 years, the fund has grown to $(40,000)$.
At that time, remove all $(40,000)$. Now, the fund has a balance of
$(0)$.
Now, re-invest $(10,000)$. After 6 years (i.e. an additional 3 years)
the fund has again grown to $(40,000)$. Again, remove all $(40,000)$. Now
the fund again has a balance of $(0)$.
Now, again re-invest $(10,000)$. After 9 years (i.e. an additional 3 years
after you cleaned out the fund the 2nd time) the fund the fund has again grown
to $(40,000)$. Again, remove all $(40,000)$. Now
the fund again has a balance of $(0)$.
The reason that I am interpreting your original posting in this manner is
that in your original posting, you use the following math expression:
$$[120,000 - 30,000] \div 30,000 = 300\%.$$
I assume from this two things:
After 9 years, you now have $120,000$, rather than (for example)
$(4^3 \times 10,000 = 640,000)$.
You have made three separate investments of $(10,000)$. The 1st
investment at year-$0$, the 2nd investment at year-$3$, and the
3rd investment at year-$6$.
What this means is that at year-$3$, when you removed the $(40,000)$
from the fund, that money never had the chance to continue to
generate interest.
Therefore, your calculation of
$\displaystyle (1 + 3)^{(1/9)} - 1 \approx 0.1665$ is inappropriate.
Assume that you invest $(1)$ for $n$ years, leaving all of the money in the fund,
with no deposits or withdrawals. Further assume that after $n$ years, your
fund has increased from $(1)$ to $(D + 1),$ for a profit of $(D)$.
Then, the formula for ROI is
$\displaystyle \left[(1 + D)^{(1/n)} - 1\right].$
However, in your original question, you (in effect) terminated the fund
after 3 years, when you withdrew all of the money. This money that you
withdrew never had the chance to continue to earn interest. Then, you
re-started the fund at year-$3$. This analysis of your actions is the
only way to explain your total accumulation of $(120,000)$, after 9 years.
That is, you must have accumulated $(40,000)$ 3 separate times.
Therefore, you have 3 separate transactions, where the appropriate ROI formula
for each transaction is
$\displaystyle \left[(1 + 3)^{(1/3)} - 1\right].$
So:
$(4^3 \times 10,000)$ is not the total return.
By terminating/restarting the fund (twice), you prevent the interest earned
in the first 3 years from continuing to earn interest and you prevent the interest earned
in the second 3 years from continuing to earn interest.
