First off, bottom line: If you have Daedric Smithing, the max bow you can make will be about 60% stronger than the Nightingale Bow. If you have any Smithing perk, via the Alchemy-Enchanting-Smithing loop you can make that type of bow stronger than the Nightingale Bow simply because the Nightingale Bow is unaffected by Smithing perks (this is true for any Smithing value over 129; max is about 533).
Everyone talks about how armor can be improved infinitely using the Alchemy-Enchant-Smithing loop (thus it's possible to have hide armor that's as strong as dragonscale).
Armor is affected by a cap, so once you get above 567 (80% damage reduction) no amount of improvement matters. That's why weaker armors can effectively be the same as Daedric/Dragonplate/Dragonscale. Weapons have no such cap.
Using the Alchemy-Enchanting loop, you can indeed improve the Nightingale Bow drastically, however, keep in mind that it is not affected by any perks. Although the best version has the same base damage (18) as a Daedric Bow, if you're taking perks in things like Daedric Smithing, the latter will be significantly stronger.
It's impossible to predict how "strong" it will appear in your inventory because that calculation depends on factors like Archery skill level and perks.
With the loop, by the way, I've been able to create a Potion of Fortify Smithing +147% and 4 pieces of Fortify Smithing +29%. I have the Smithing perk tree filled out going the Heavy route, so I have Daedric Smithing. I then improved my Daedric Bow to where it's doing 599 damage for me (a bunch of Fortify Archery +48%, nearly full Archery perk tree). When I did the same for the Nightingale Bow, of which I have the best version, I get to 379 damage. So the max Nightingale Bow will be around 37% weaker than the max Daedric Bow.
Just to think about the math for a second, I believe that gets me to
smithing = base skill * (1 + alchemy fort) * (1 + enchanting fort)
= 100 * (1 + 1.47) * (1 + 0.29 * 4)
Plugging that into improvement with/without the Smithing perk (Daedric Bow = Daedric Smithing, Nightingale Bow = none/without):
with = floor((smithing - 5) / (103 / 6)) * 1.8
= floor(30.76) * 1.8
without = floor((smithing + 11 / 3) / (103 / 3)) * 1.8
= floor(15.6) * 1.8
Since the base for both is 18, that brings us to 72 for the Daedric Bow and 45 for the Nightingale Bow, or 37.5% less (and the ratio stays the same regardless of skill level/perks). Note also that the above 27 damage difference is greater than the difference between the weakest and strong base damages for bows, therefore the strongest bow will be the best bow type for which you have a Smithing perk. This should always be true if you can buff Smithing over 129 using enchanting/alchemy.
Also, FWIW, by the time you have 100 Enchanting you really should take the Extra Effect perk. That means the bow above can have the same effects as the Nightingale Bow. Not that they matter much when you're one-shotting everything.