The sheer number of weapons in Elden Ring dwarfs that of FromSoftware's previous games. But if you want these weapons to be effective against the higher level enemies you will face in the furthest reaches of The Lands Between, you will need Runes, as well as Smithing Stones which are a little harder to come by.
Though you can find and eventually buy lower level Smithing Stones, the max level upgrade for normal weapons - level 25 - requires the special Ancient Dragon Smithing Stone. Unfortunately, there are limited quantities in each run, but fortunately for you, this list will tell you where to find all the Ancient Dragon Smithing Stones in Elden Ring.
13 Nepheli Loux Quest Line
The Ancient Dragon Smithing stone that is probably the first you can acquire in Elden Ring is received after completing Nepheli Loux's Quest line and then meeting her at the throne room of Stormveil Castle. This is just a little ways past the site of grace in the Godrick the Grafted boss arena
It should be noted that this quest also requires you to complete the Kenneth Haight quest as well, as it is related to her quest being completed.
12 Gatekeeper Gostoc (After Completing Nepheli Loux Quest Line)
This Ancient Dragon Smithing Stone is sold to you by Gatekeeper Gostoc, who you may have met earlier while progressing through Stormveil Castle. After completing Nepheli Loux's quest, Gatekeeper Gostoc should be to her side selling things.
However, you can now purchase an Ancient Dragon Smithing Stone from him for 20,000 Runes. Keep in mind that he must still be alive for this to work.
11 Sorceress Sellen Quest Line
Sorceress Sellen is one of the NPC's in Elden Ring that is vital to meet for any aspiring spell caster as she sells a great deal of useful sorceries. Completing her quest also rewards you with rare and valuable spells, but also another Ancient Dragon Smithing Stone.
Like with the Gostoc Stone, there are ways to miss this Stone so be careful to side with Witch-Hunter Jerren at the final step of Sorceress Sellen's quest.
10 Gurranq Quest Line
While exploring the Lands Between you may find yourself a Tibia Mariner, one of the Field Bosses in Elden Ring, and procure yourself some Deathroot. You can take this Deathroot to the NPC Gurranq who is located in the Northwest of Caelid.
The rewards that Gurranq gives in return for Deathroot are mostly Bestial Incantations, but the ninth reward will be yet another Ancient Dragon Smithing Stone.
9 Mountaintops of the Giants
Once you reach the Mountaintops of the Giants, having a highly upgraded weapon will be very important as the enemies in this area are very powerful. Luckily, there is another Ancient Dragon Smithing Stone here.
From the Church of Repose Site of Grace, look to the southeast to spot a massive skull covered in snow. You will also spot several dangerous enemies, but make your way into the mouth of the skull without dying to retrieve your prize.
8 Crumbling Farum Azula: First Dragon Encounter
Getting this far in Elden Ring is no easy feat, and the game will reward your hard fought victories in Farum Azula. Almost immediately after arriving, a dragon will swoop down and begin attacking you on a slanted platform right.
This foe is similar in move set to the Lichdragon, so expect a challenge. Though, challenging may be putting it lightly, defeat this difficult dragon to receive an Ancient Dragon Smithing Stone.
7 Crumbling Farum Azula: Dragon Temple
This Ancient Dragon Smithing Stone has a different challenge tied to getting it. Instead of being a reward from an NPC or from defeating a tough foe, this Ancient Dragon Smithing Stone is simply well hidden in Farum Azula.
To find it, start at the Dragon Temple and progress up to a spiral staircase. After heading up, find a floating pillar towards the south and follow the path it points out to find the Smithing Stone.
6 Crumbling Farum Azula: Dragon Temple Dragon Encounter
Dragons are all over Farum Azula, and some of them are definitely worth fighting for the rewards they drop upon death. This specific dragon is another of the Lichdragon variety and can be found close to the Dragon Temple Rooftop Site of Grace.
Progressing forward from here, a dragon will rain lightning attacks on you from afar. Make your way to it and defeat it to receive another valuable Ancient Dragon Smithing Stone.
5 Consecrated Snowfield
If you have made it all the way to the Consecrated Snowfield, you will know how strong the enemies are at this point in the game. To give you an advantage in encounters with these foes, fully upgrading weapons is a must.
And luckily, there are no enemies that have to be killed to get this Ancient Dragon Smithing Stone. It is closest to the Inner Consecrated Snowfield Site of Grace. Just follow the river to the east, and dodge past the Magma Wyrm to get this Smithing Stone.
4 Consecrated Snowfield Night's Calvary
Also near the Inner Consecrated Snowfield Site of Grace is a caravan pulled by giants with a smattering of enemies around it. At night, however, two of the Night's Calvary field bosses will spawn to defend it.
Successfully felling both of these bosses without dying or resting at a site of grace will reward you with the Night's Calavary armor set and an Ancient Dragon Smithing Stone.
3 Yelough Anix Tunnel
Within the Yelough Anix Tunnel, which is located in the Consecrated Snowfield on the west side, you can find another Ancient Dragon Smithing Stone. This is another one that is just a pickup on a body.
It is not tied to killing any enemies, though the tunnel itself has a few pretty dangerous ones. After progressing a ways in, make sure to drop down to collect this Smithing Stone but watch out for the enemy that waits near it.
2 Miquella's Haligtree
Miquella's Haligtree is one of the final areas in Elden Ring, and it definitely lives up to the past games in that regard. Souls games are infamous for only getting harder over time and Miquella's Haligtree is no exception.
In the Haligtree Town area, find a statue guarded by a Lesser Leonine Misbegotten enemy and two regular Misbegotten enemies. The Ancient Dragon Smithing Stone is in front of the statue nearby.
1 Miquella's Haligtree (After Loretta Boss Fight)
Progressing through Miquella's Haligtree will eventually pit you against Loretta, Knight of the Haligtree. While she does put up a fight, dispatching her should be no problem if you have been using the previous upgrade materials wisely.
After you defeat Loretta, take the ladder ahead and instead of riding the elevator at the bottom, climb the stairs nearby to find a chest with the final Ancient Dragon Smithing Stone in Elden Ring.