Is there any way to understand the magnitude of
$$10 \uparrow \uparrow \uparrow 10$$
?
I know that the number can be constructed as follows :
$$M_1 := 10$$
$$M_2 := 10^{10^{10^{...10}}}$$
where the height of the power tower is $M_1 = 10$
$$M_3 := 10^{10^{10^{...10}}}$$
where the height of the power tower is $M_2 = 10\uparrow \uparrow 10$
$$M_4 := 10^{10^{10^{...10}}}$$
where the height of the power tower is $M_3$
and so on
Then, $M_{10}$ is the desired number.
But since the heights of the power towers are too big to be understood, it seems hopeless to get any idea of the magnitude of $M_{10}$. Am I right ?
The magnitude of $10↑↑↑10$ can be understood, but not with ordinary power towers.
There are at least two things you can do to make such number more accessible:
(1) Use powerful and relevant notation.
For a simple (but rough) start, you can create a list of milestone numbers using the notation $10↑^{k}n$, with $2≤n≤9$. Note that $10↑^{k}10=10↑^{k+1}2$, so whenever n increases from 9 to 10, you can drop it back to 2 and add another arrow.
This way you'll have a continuous progression of numbers, starting with ordinary scientific notation:
$10↑2 = 10 \times 10 = 100$
$10↑3 = 10 \times 10 \times 10 = 1000$
$10↑4 = 10 \times 10 \times 10 \times 10 = 10{,}000$
$10↑5 = 10 \times 10 \times 10 \times 10 \times 10 = 100{,}000$
$10↑6 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1{,}000{,}000$
$10↑7 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10{,}000{,}000$
$10↑8 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 100{,}000{,}000$
$10↑9 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1{,}000{,}000{,}000$
$10↑10 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10{,}000{,}000{,}000 = 10↑↑2$
Continuing with the simple power towers:
$10↑↑2 = 10^{10}$
$10↑↑3 = 10^{10^{10}}$
$10↑↑4 = 10^{10^{10^{10}}}$
.
.
.
$10↑↑10 = 10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}} = 10↑↑↑2$
And finally, the iterated power towers:
$10↑↑↑2 = 10↑↑10$
$10↑↑↑3 = 10↑↑10↑↑10$
$10↑↑↑4 = 10↑↑10↑↑10↑↑10$
.
.
.
$10↑↑↑10 = 10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10 = 10↑↑↑↑2$
That's the number scale I recommend you use, to "reach" $10↑↑↑10$ without overwhelming yourself. The important thing here is, that by the time you add the third arrow, you aren't working with the actual power-towers anymore. At this point you should forget them altogether, and think solely in terms of double-arrows. Just like you wouldn't bother trying to build a googolplex by multiplying a googol $10$'s together.
(2) Familiarize yourself with some actual uses of these numbers.
It is very difficult to understand the magnitude of a number, when one doesn't have a few examples of what this number actually means. We understand millions and billions because we have examples of things that these numbers refer to, but what could a number like $10↑↑↑10$ actually be compared to?
Fortunately, numbers of this magnitude appear in quite a few areas of mathematics. Besides the obvious "application" in recreational mathematics, such numbers appear naturally in more serious topics such as Goodstein Sequences, Turing Machines, Ramsey Theory and even when dealing with very foundation of mathematics itself. Familiarizing yourself with such examples will make the numbers much more "real" to you.
For example, the 5th Goodstein Sequence terminates after roughly $10↑↑↑4$ terms. The 6th one terminates after roughly $10↑^56$ terms. So your $10↑↑↑10$ is somewhere in between the lengths of these two sequences. Once you know what a Goodstein Sequence actually is (and after you play with them a bit), these facts really do help you understand the magnitude of the numbers involved.