16 14 or 12 bytes
The only native integer literals supported by Befunge are the numbers 0 to 9, so coming up with an efficient representation for larger values is actually quite a common problem. In this case, though, it's not just a matter of finding the most efficient representation for a number, but also choosing a target number whose best representation is the shortest of all possible targets.
After trying thousands of different combinations, the best I've been able to do so far is 14 bytes (12 to calculate the number, and then two more bytes to output it).
Try it online!
If we allow extended ASCII characters, though, we can improve on the above solution by two bytes with:
This is using the Latin-1 encoding, which we unfortunately can't demonstrate on TIO, since it doesn't support 8-byte encodings for the Befunge interpreters.
In the first answer, we start with the string
"m&", which essentially pushes the values 109 and 38 onto the stack. Note that we save a byte by dropping the opening quote, since we can rely on Befunge's wrap-around playfield to have the one quote both open and close the string (this pushes a number of additional values onto the stack that we don't need, but we only care about the final two).
We then push an additional
7 onto the stack, and execute the sequence
*:*::**+, which performs a set of arithmetic calculations with the stack values, essentially calculating the expression:
109+(38*7)^2^3 = 354233654641325
Finally we use the
. instruction to output the result of the calculation, and then the
@ instruction ends the program.
In the second answer, we start with the string
"eûãÃ", which pushes the values 101, 251, 227 and 195. We then execute the sequence
**:*+, which performs the required calculations on those values:
101+(251*227*195)^2 = 123443543565326
And again we finish with the instructions
@ to output the result and exit.
Note that in both cases you need to be using an interpreter which supports 64-bit integers. Also note that the string wrapping trick won't work on Befunge-98 interpreters, since it relies on the interpreter ignoring instructions it doesn't recognise, but in Befunge-98 an invalid instruction will reflect.