It would be awesome for say Shadowrun game system to allow us the option to have "exploding dice".

in older versions it did, in 4th edition it has something similar when you use edge if i remember correctly.  Otherwise 4th ed SR (and i'm assuming 5th) has no need for exploding dice since it counts hits on the dice you roll and those are always 5's and 6's

SR5 rolls explode when you use Edge to boost a roll, and the dice roller does handle it; it's the "Reroll Max" option.

Is there a way to restrict the explosions? For instance, in PF there's a class feature for Swashbucklers which adds a d6 to skill checks, but can explode up to <Dex Mod> times.

If the record each die records each explosion, then that might be at least a potential, if imperfect, solution.

Gotta give this my +1.