Pretty much every d20-game uses a d20 roll to determine initiative at the beginning of an encounter. Also, pretty much every d20-game has some instances in which a natural 20 gets a special treatment (e.g. critical hits, auto-success).

Are there notable interesting examples of games featuring a special benefit of a natural 20 for initiative? What is this benefit? Are there common house rules in games without such a feature?

Some thoughts to explain where I am coming from with this question: As far as I am aware, no such rules exist for the games I have played so far (mostly DnD 3.5 and upwards). This question plays with the idea of using the card drawing initiative from Savage Worlds in DnD 5e, and I find the aspect of the joker particularly interesting. If initiative is rolled separately for every round, granting advantage on ability checks, saving throws and attack rolls for that round does sound like a nice little unpredictable factor in an encounter. However, if initiative is fixed for the entire encounter, this would surely be over-powered. I still like the general idea, so I am looking for similar rules and rulings that might inspire a good house rule.

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    \$\begingroup\$ What would make one answer to this question better than another? That is, despite the narrowness of the request, this is currently just an opportunity for discussion, and discussions are better suited to a forum. I suspect, as presented, this will likely to be put on hold as off-topic for the site. \$\endgroup\$ – Hey I Can Chan Jun 24 '19 at 17:28
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    \$\begingroup\$ It's also a close match to #2 on the don't ask list, unfortunately. It's only missing the "I do this" part. \$\endgroup\$ – JohnP Jun 24 '19 at 17:31
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    \$\begingroup\$ (Keep in mind that the site knows what you want, but it can't provide it; it's just not supported by the site's format. Instead, for example, you can detail the the house rule you're considering and ask if it's balanced then let folks weigh in with their experiences with a similar house rule… or with their experiences with a printed rule from another game that's like the proposed house rule. Or, like the other question, ask how transferring a mechanic from one RPG will work in another. However, the site won't generate ideas so that those questions can then be asked. I know, right? :-)) \$\endgroup\$ – Hey I Can Chan Jun 24 '19 at 17:47
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    \$\begingroup\$ @gatherer818 Existence questions are borderline. \$\endgroup\$ – Hey I Can Chan Jun 24 '19 at 17:47