The Prime Bubble rolls a 6, meaning that the Cascader will iterate six times

#1: We always start with a 6 sided die, and it rolls a 2, so the next die has 6x2=12 sides
#2: The 12 sided die rolls an 8, meaning that the third die has 12x8=96 sides
#3: The 96 sided die rolls a 35, meaning that die 4 has 96x35=3360 sides
#4: The 3360 sided die rolls a 2290, so die 5 has 3360x2290 = 9,817,920 sides
#5: The 9.8 million sided die rolls a 5,101,894, so the final die has 50,090,870,753,280 sides
#6: The 50 trillion sided die rolls a one. Hooray.
Since the last die rolled gave a 1, your function or program should output 1.

The Prime Bubble rolls a 2, meaning that the Cascader will iterate twice.

#1: We always start with a 6 sided die, and it rolls a 4, so the next die has 6x4 = 24 sides
#2: The 24 sided die rolls a 14

Since the last die rolled gave a 14, your function or program should output 14.

(*1+?)^:(?`])@6

*|{x,1+1?x:(*).x}/[*a;6,a:1+1?6]

*|{x,1+1?x:(*).x}/[*a;6,a:1+1?6] / the solution
  {             }/[n;    c     ] / iterate over lambda n times with starting condition c
                            1?6  / 1 choose 6, between 0..5 (returns a list of 1 item)
                          1+     / add 1 (so between 1..6)
                        a:       / store as 'a'
                      6,         / prepend 6, the number of sides of the first dice
                   *a            / we are iterating between 0 and 5 times, take first (*)
           (*).x                 / multi-argument apply (.) multiply (*) to x, e.g. 6*2 => 12
         x:                      / save that as 'x'
       1?                        / 1 choose x, between 0..x-1
     1+                          / add 1 (so between 1..x)
   x,                            / prepend x
*|                               / reverse-first aka 'last'

(6 3        / 1 choose 6 => 3, so perform 3 iterations
 18 15      / 1 choose (6*3=18) => 15
 270 31     / 1 choose (18*15=270) => 31
 8370 5280) / 1 choose (270*31=8730) => 5280

X6DLΩF*DLΩ

6×X$6X’¤¡X

¤   | Following as a nilad:
    6X     | - A random number between 1 and 6
      ’    | - Decrease by 1 (call this N)
6          | Now, start with 6
   $    ¡  | Repeat the following N times, as a monad
 ×         | - Multiply by:
  X        |   - A random number between 1 and the current total
         X | Finally, generate a random number between 1 and the output of the above loop

(r=RandomInteger)@{1,Nest[r@{1,#}#&,6,r@5]}

6X×$5СXX

6X×$5СXX - Link: no arguments
6         - initialise left argument to 6
    5С   - repeat five times, collecting up as we go: -> a list of 6 possible dice sizes
   $      -   last two links as a monad = f(v):           e.g [6,18,288,4032,1382976,216315425088]
 X        -     random number in [1,v]                     or [6,6,6,6,6,6]
  ×       -     multiply (by v)                            or [6,36,1296,1679616,2821109907456,7958661109946400884391936]
       X  - random choice (since the input is now a list) -> faces (on final die)
        X - random number in [1,faces]

⊞υ⁶Ｆ⊕‽⁶⊞υ⊕‽ΠυＩ⊟υ

⊞υ⁶

Ｆ⊕‽⁶

⊞υ⊕‽Πυ

Ｉ⊟υ

≔⁶θＦ‽⁶≧×⊕‽θθＩ⊕‽θ

≔⁶θ

Ｆ‽⁶

≧×⊕‽θθ

Ｉ⊕‽θ