OOC:
No that's not quite right. The way you work out your kept dice pool is as follows:
1) Roll your Skill and Ring Dice as appropriate for your Skill and Approach
2) Add any other dice to your pool from any other abilities (such as 'Blood of the Kami')
3) Before re rolling any exploding dice work out which dice you are keeping from your initial roll plus extra dice. The maximum number of dice you can usually keep at this point is equal to the value of the Ring you used in the check but can be amended for other factors (i.e. Certain school abilities, spent Void Points, etc)
4) Once you have worked out your Kept dice pool roll any extra dice for any Explosive Successes in your Kept pool (you do not re roll any that you did not keep)
5) You can decide to add any of those to your Kept pool even if that means your number of Kept dice exceeds your Ring value. If any of the additional dice you keep also roll Explosive successes you can roll additional dice for those as well and, dependent on the results, choose whether to add them to your pool of kept dice.
So in the example of your roll you got the following:
Ring Dice: 3 (Opportunity), 4 (Success and Strife), 6 (Explosive Success, Strife)
Skill Dice: 5 (Opportunity)
Breath of the Kami Dice: * (Opportunity)
So out of the above you can keep 3 dice, if one of the dice you keep is the Explosive Success then you can roll an additional die and add it to your kept pool if you want to even if that means you exceed 3 kept dice.
One other thing to remember is you only need to keep a minimum of 1 dice, can be useful in situations where you roll lots of Strife and a Blank die and don't want to risk unmasking.
Does that clear it all up? Just let me know how you're managing your dice pool for the next check and I'll redo the post with the result of it before moving on.
OOC:
Yeah they just keep exploding but as noted above you only re roll for an exploding die if you are keeping it and you are also not under any obligation to keep any of the extra dice you roll as part of keeping an explosive success.