Diffie-Hellman works as follows:
Given public parameters $p$ (a large prime) and $g$ (always referred to as a generator of $(\mathbb{Z}^∗_p)$. Then:
Alice randomly chooses $a<p$ and sends $A\leftarrow g^a \mod p$ to Bob;
Bob randomly chooses $b<p$ and sends $B\leftarrow g^b \mod p $ to Alice;
Alice computes $S\leftarrow B^a \mod p$;
Bob computes $S\leftarrow A^b \mod p$.
What happens if we choose $a$ and $b$ grater than $p$?
The modulus operation $\pmod p$ is performed at each step and reduces the result into $\bmod p$. And a clever implementation can use the Fermat's Little Theorem instead of taking the power than reducing to modulo $p$. After that it's possible to use the modular version of repeated squares algorithm or similar.
Example 1) code used from sublimerobots
sharedPrime = 23 # p
sharedBase = 5 # g
aliceSecret = 600 # a
bobSecret = 1500 # b
Alice Sends Over Public Chanel: 8
Bob Sends Over Public Chanel: 4
Privately Calculated Shared Secret:
Alice Shared Secret: 2
Bob Shared Secret: 2
Example 2)
sharedPrime = 23 # p
sharedBase = 5 # g
aliceSecret = 6000000 # a
bobSecret = 15000000 # b
Alice Sends Over Public Chanel: 8
Bob Sends Over Public Chanel: 4
Privately Calculated Shared Secret:
Alice Shared Secret: 2
Bob Shared Secret: 2
I think you are confusing the mathematical representation and the actual value.
In the sense of optimization, the code from sublimerobots is not good. Actually. instead of
bobSharedSecret = (A**bobSecret) % sharedPrime
a faster version
bobSharedSecret = pow(A,bobSecret,sharedPrime)
which uses modular binary exponentiation.
It could have three consequences.
1) If you are very unlucky, and you pick a "zero" ($a$ such that $a=0\mod p-1 $), it will break your system : (but this will happen with a negligible probability, and it could be detected) An external observer will easily guess the shared secret
2) You lose in efficiency
3) Your integer had to be chosen upper-bounded (you can not pick uniformly over all the integers, if you choose badly this point (something not divisible by $p$), it will create a bias in the distribution of your keys (probably not a problem in practice, but in theory it's less secure).
