I've played with this idea in the past. Here are a few things I played around with to address some of the drawbacks:
1.) Instead of rolling more dice, you just add the average of the die to the result (rounded up). Effectively, a D6 would have sides of 1, 2, 3, 4, 5 and 10. A d12 would have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 19. WHY: Prevent a second or third round of die rolling.
2.) An ALTERNATIVE that does not work with 1: Subtract 1 from the result once you have rolled the maximum number. WHY: This allows a d6 roll to still end up as a 6. Example: I roll a d6 and get a 6. I explode for 1 more d6 and get a 1. My result is 6 + 1 - 1 ... 6.
Damage rerolling mechanics gain power in this situation. For example, Savage Attacker is more powerful. This isn't necessarily a problem, but you do want to be aware of it.
You need to be aware that this is going to be
actually be more deadly. I know that is the goal, but be aware that it will truly work. While people will focus on the average die rolls when figuring out the impact of this, that approach overlooks that we don't always get the average, and the most impactful situations will be far from average. It is those fringe situations that will kill PCs, and perhaps result in a TPK. And they're not that fringe.
Take, for example, a CR 5 Hill Giant. A group of 5 3rd level PCs that encounter a Hill Giant in the woods might be facing a 'hard' encounter. Not even deadly.
However, when that Hill Giant throws a rock, they get 3 shots at a 10 on a d10. They normally deal an average of 21 damage, or ~23 under this system. However, if we fix one of those dice as a 10, that average damage becomes ~33. And this is just average damage - it can be significantly higher. Those numbers are enough to easily take a pc down in one shot - and perhaps beyond their negative hp max, and are not all that unlikely. Go ahead and roll 20 different hill giant rocks and see what you get. One of them was likely to be horrifically high. Picture that blow coming at the start of a combat with the hill giant, and then seeing how the PCs are able to repond down one PC from the start. That is always a possibility, but it is a lot more likely in this type of scenario.
This would move that range where PCs die - a lot - from levels 1 to 2 all the way up to levels 5 or 6. You will have costly revivy available, perhaps, then, but it will be needed more often.
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EDIT: in case you are interested in the math, it bumps the average by N/(N-1). So, if your d6 averages 3.5, exploding the die (without limit) increases the average by 6/5, bringing it up to 4.2.
6/5 is 1.2, which added to a normal 3.5 would be 4.7.