How to estimate time to reach X with an exponentially increasing growth rate I need to estimate the amount of time it will take to reach a desired number of X, when it's growth rate is constantly increasing (exponentially?).
Example
In this example, there are 4 layers, with each layer from bottom to top feeding into the layer above it at a constant time interval, and finally the top layer which produces the final output X. A: producing A of X every 1 second
B: producing B of A every 2 seconds
C: producing C of B every 3 seconds
D: producing D of C every 4 seconds

With this, we know that C is increasing at a rate of D every 4 seconds, B is increasing at a rate of C every 3 seconds, and so on up the chain to produce the final output of A of X every 1 second.
Goal
Let's say I currently have 1,000 X and our goal is to reach 10,000 X. How can I estimate the amount of time t it will take to reach 10,000 X with an arbitrary number of layers feeding into each other?
I need the final equation to be able to work with any number of layers and variances in each layer's individual time interval. I only included 4 layers and easy output numbers for the purpose of a simple example.
Descriptive Example
For better demonstration purposes, a more specific example with exact outputs from zero to 4 seconds below.
Initial state (0 seconds):

X: 0
A (100): producing 100 of X every 1 second
B (50): producing 50 of A every 2 seconds
C (25): producing 25 of B every 3 seconds
D (10): producing 10 of C every 4 seconds

After 1 second:

X: 100 (+100)
A (100): producing 100 of X every 1 second (produced 100 X)
B (50): producing 50 of A every 2 seconds
C (25): producing 25 of B every 3 seconds
D (10): producing 10 of C every 4 seconds

After 2 seconds:

X: 250
A (150): producing 150 of X every 1 second (produced 150 X)
B (50): producing 50 of A every 2 seconds (produced 50 A)
C (25): producing 25 of B every 3 seconds
D (10): producing 10 of C every 4 seconds

After 3 seconds:

X: 400
A (150): producing 150 of X every 1 second (produced 150 X)
B (75): producing 75 of A every 2 seconds
C (25): producing 25 of B every 3 seconds (produced 25 B)
D (10): producing 10 of C every 4 seconds

After 4 seconds:

X: 625
A (225): producing 225 of X every 1 second (produced 225 X)
B (75): producing 75 of A every 2 seconds (produced 75 A)
C (35): producing 35 of B every 3 seconds
D (10): producing 10 of C every 4 seconds (produced 10 C)

A: You can just follow up the chain.  Each thing you start with generates $X$ in a given pattern.  The situation is linear, so you can just add things up.  The number of $D$ never changes, so $D(t)=D(0)$.  Since each $D$ makes a $C$ after $4$ seconds, it makes $\lfloor \frac t4\rfloor \ C's$ after $t$ seconds and $C(t)=\lfloor \frac t4\rfloor D(0)$ because we will count the $C'$s that were at the start later.  All those $C$'s make $B$'s after $3$ seconds, so $B(t)=B(t-1)+C(t-3)=B(t-1)+\lfloor \frac {t-3}4\rfloor D(0)$.  That equation hides the fact that the number of $B$'s is quadratically increasing in time because the second term is linearly increasing in time.  Then we have $A(t)=A(t-1)+B(t-2)=A(t-1)+B(t-3)+\lfloor \frac{t-5}4\rfloor D(0)$  Now it is the $B(t-3)$ term that makes $A(t)$ cubic in $t$.  Finally, $X(t)=X(t-1)+A(t-1)=X(t-1)+A(t-2)+B(t-3)$ is quartic in $t$.
