I would cast Wish that you were my DM

Wish is the mightiest spell a mortal creature can cast. By simply speaking aloud, you can alter the very foundations of reality in accord with your desires.

The basic use of this spell is to duplicate any other spell of 8th level or lower. You don't need to meet any requirements in that spell, including costly components. The spell simply takes effect.

Alternatively, you can create one of the following effects of your choice:

• You create one object of up to 25,000 gp in value that isn't a magic item. The object can be no more than 300 feet in any dimension, and it appears in an unoccupied space you can see on the ground.
• You allow up to twenty creatures that you can see to regain all hit points, and you end all effects on them described in the greater restoration spell.
• You grant up to ten creatures that you can see resistance to a damage type you choose.
• You grant up to ten creatures you can see immunity to a single spell or other magical effect for 8 hours. For instance, you could make yourself and all your companions immune to a lich's life drain attack.
• You undo a single recent event by forcing a reroll of any roll made within the last round (including your last turn). Reality reshapes itself to accommodate the new result. For example, a wish spell could undo an opponent's successful save, a foe's critical hit, or a friend's failed save. You can force the reroll to be made with advantage or disadvantage, and you can choose whether to use the reroll or the original roll.

If you attempt to do one of these effects, the Wish is guaranteed to work as intended. Everything else is up to your DM.

• Your DM might say Wishes are god status and let you wish for more wishes, destroy planets, make the whole multiverse into an egg, etc.
• Your DM also might say that Wishes only work as written, causing your wish to be wasted if it goes beyond that.
• Your DM also might take a more balanced approach and consider very carefully what to have happen based on the needs of your campaign, the wording of your wish, and their own take on the limitations of the Wish spell.

All of those situations are solidly within the RAW and RAI. Your DM is designed to have whatever latitude they want in this, mechanically and thematically. Your DM has tools, such as exploiting wording, causing the wish to fail, or causing unintended consequences not based on wording, to deal with Wishes that will break their game.

So yes, you might be able to wish for more Rings of Three Wishes, or any other thing that would grant you more and/or infinite wishes. You might not. Either way, you have the same chance to suffer the stress of casting Wish, and either way, your DM has final say in what happens.

Remember it is the job of everyone at the table, not just the DM and not just the players, to make sure everyone has fun. Don't wish for something to intentionally sabotage your game, and DMs don't veto wishes just because you don't like what the player wished for. Have fun and be fair to each other. Have fun out there guys :D

PS. If you're a DM and unsure how to combat players exploiting wishing for stuff like this, here are my favorites:

• "I wish for more rings of 3 Wishes", etc: "A pile of rings identical to the one you now wear appears at your feet. None of them have any charges." (alt. "They are all nonmagical rings that are only visually identical to yours", etc)
• "I wish for the charges to refill on my ring of 3 wishes", etc: "Your ring refills on charges, but may now only replicate the effects of an 8th level spell or lower." (alt. "The gems light up, appearing to refill on charges, but none are actually refilled", etc)
• "I wish for everything on this paper to come true", etc: "The text on your paper changes so that each statement is true" (eg. "The Sky is Blue") (alt. "Everything on your paper is true. You do wish you had those things." etc.)

Adding this to my campaign 100%

If a player used a wish spell to wish for more wishes, I would say, "How many do you want?" I would encourage them to be greedy.

"I want infinite wishes!"

I then would look crestfallen and ask, "Are you sure?"

If the player falls for it, I would pick up my stuff and move to the player's seat, gesturing for them to take  my spot. "Congratulations, you are now the DM."

Quote from Kirtical_Mass >>

• "I wish for more rings of 3 Wishes", etc: "A pile of rings identical to the one you now wear appears at your feet. None of them have any charges." (alt. "They are all nonmagical rings that are only visually identical to yours", etc)

Just crush them in an avalanche of rings.

My DM gave me this item and I really want to make him regret that so can I technically wish for the charges to be replenished with the last charge? 

Actually, for the spell wish, you aren't able to wish for any magical items. the ring is in fact, a magical item