Assuming as standard the set: d4, d6, d8, d10, d12, and d20, all sequentially numbered from 1 to number of sides.

Dungeon Crawl Classics also uses d3, d5, d7, d14, d16, d24, d30, and d%.

Many older games use half dice: d2, d3, d5. AD&D, the Arcanum, Hero System, RuneQuest, DragonQuest, Dark Heresy, Deathwatch, Rogue Trader, all the 1pg system games, and many others.

Traveller: the New Era used 1d16 for aging saves, but faked it with a d20, rerolling 16-19. (It used d6, d10, & d20 otherwise.)

Fudge and Fate systems normally use rolls of 4dF to generate a range from +4 to -4. Each dF is marked plus, plus, blank, blank, minus, minus. Diaspora, Dresden Files, Spirit of the Century, and several others use dF as standard, and several others use them as options.

Hero System and some Fuzion system games use "body dice" as an optional accessory. They include standard numbers, but also dots... 1 dot on 2-5, 2 on 6, no dot on 1. When told to count the body, count the dots.

Burning Wheel, Burning Empires, Mouse Guard, and d6 Batman all use d6, but custom marked dice are available to speed play. Mouse Guard dice are marked Axe, crossed sword x2, and snake x3. Burning Wheel black dice have special success mark, success mark, fail mark, and special fail mark; black has 2 success and 2 fail, grey 3 and 1, white 4 and 0, but the specific symbols vary by run. Batman dice have 4 bat-symbols, and 2 jokers; one of the dice in the set is off color, and has a special mark on one joker and one bat sign.

The One Ring uses special marked d6 and d12. The d12 is marked 1-10, sauron, gandalf. The d6's are numbered 1-6, but 1-3 are in outline text, and 6 has a small rune next to the 6. Standard dice can be used with little extra effort.

d7, d24, d30, all have ready use in time randomisation. d28 are used in a 13 month calendar.

Many board games also use non-standard dice. Some need to be watched for in competitive play...

Formula De and Formula D use standard looking d4, d6, d8, d12, d20, and d30, but they are not standard numbered. They are, however, numbered.

There's a built-in explode function, but you have to use it before doing your other processing. Specifically, use FOOdBAR to have BAR evaluated as a sequence and used to generate FOO "dice" with arbitrary side counts and values. Here, DICEd([explode d6]) does the job, turning an exploded d6 into the base for the actual roll. (explode doesn't separate dice that you give it — you have to do that, or you'll get a 5d6 roll that only explodes, once, on 30.)

You can't compare sequences to a threshold to get a list or sum of values against that threshold — you'll just get the sequence summed and checked against the threshold. Instead, use [count VALUES in SEQUENCE] for each of the valid values (3-6). And as it turns out, since every 6 triggers a new die and all those dice count, you can just double-count 6s once exploded. So really that should be [count {3..6, 6} in ROLL].

The results, as best I can manage, aren't pretty. Each explosion will either fumble harder (1), make no progress against fumbling (2-5), or break even and get a new chance to do the same thing (6), and the more dice you start with, the more chances for explosions. There is no provision for fumbles to peter out in explosions, as one would expect from a great success; the only question is whether the chain ends with a 1 or not.