The crux of the question revolves around what chasing a high AC actually accomplishes. Without knowing the end goal, it's impossible to judge whether or not you've fulfilled your aims. This answer is going to proceed under the assumption that you're just trying not to get physically hit the majority of the time rather than completely mitigate your opponent's offense. The latter is better accomplished by utilizing miss chances and conditions.
A good metric for "relevant" would be your average opponent needing an 11 or better to hit you, leaving aside circumstantial effects like flanking, cover, etc. A "high" AC would be closer to needing a 15+ to hit you, so we'll use both of these as our guideposts.
A survey of all the CR 1 monsters in the SRD yields an average attack bonus of +2.7 with two outliers at +5 (small air/earth elemental) and +6 (grig). Using our above targets, that suggests an AC of 14 to 18, with an absolute cap of 21 for the ultra-cautious. The vast majority sit right at +2 to +3, so shooting for average is perfectly safe at this level.
At CR 5, the average attack bonus rises to +9.7, with high outliers at +13 (Greater Barghest and Werebear hybrid/bear form). This suggests an AC of 21 to 25 with a cap at 28.
Moving on to CR 10, the average attack bonus is now +18.6, with two high outliers at +25 (Colossal Animated Object and Juvenile Red Dragon). Target ACs should be 30 to 35 with a cap of 40. At this point, the divide between attach-oriented monsters and special-ability-oriented monsters begins to get wildly apparent. The high attacks are more than 12 points higher than the low attacks. Raw AC is not likely to be the single answer it was at lower levels.
At CR 15, the field is almost exclusively dragons, which skews the average attack bonus to +26.9. Expanding to include CR 14 monsters drops the average to +24.8. The outliers are pretty much every dragon in the book at +28 to +33. Average AC target jumps to 36 to 40 with a cap of 48. A realistic gauge of attacks is hard to establish because chances are you're facing something with class levels, templates, or special abilities that don't rely on attack rolls, such as spells.
At CR 20, you're down to very powerful dragons, top-tier NPCs, pit fiends, and balors. Average attack bonus is +37.0, but swings from +30 to +46, excluding whatever overpowered BBEG your GM has in mind. AC targets are difficult to properly set due to such a small sample size, but 50 to 55 is a decent ballpark. At this stage, raw AC doesn't mean as much because you'll be facing numerous save-or-die effects, massive area effects, and other non-attack offensive abilities. Even a cheap ring of evasion will probably save your life more than cranking out that last 10 to 15 AC points.
Actually achieving any of these AC targets is an entirely different kettle of fish.
At 1st level, a decent suit of armor or an exceptional Dexterity score will get you to AC 14 to 16 easily. Add on a heavy shield and you're at AC 18 without even trying.
At 5th level, it starts requiring some effort. To hit an AC of 25:
- equip a suit of +1 full plate (+9)
- rely on a little speed (+1 Dex)
- use a +1 heavy shield (+3)
- grab the Dodge feat (+1)
- wear a +1 ring of protection
That even leaves you enough gold for a +1 weapon and a few other trinkets.
At 10th level, you're going to have to invest a significant portion of your wealth (approx. 45,000 gp) to hit your AC target of 34-39:
- +3 mithril full plate (+11)
- +3 heavy shield (+5)
- +2 ring of protection (+2)
- +2 amulet of natural armor (+2)
- 16+ Dex (+3)
- Dodge feat (+1)
- Combat Expertise feat (+0 to +5)
Above 10th, conventional AC adds are going to eventually top you out at 57 for a total cost of about 237,000 gp (half that if you can find a caster to sink about 9120 XP into crafting it):
- +5 defending weapon (72k gp) [+5]
- +5 mithril full plate (35.5k gp) [+13]
- +5 heavy steel shield (25.3k gp) [+7]
- +5 ring of protection (50k gp) [+5]
- +5 amulet of natural armor (50k gp) [+5]
- Dodge feat [+1]
- Combat Expertise feat [+5]
- fight defensively with 5+ ranks in Tumble [+3]
- 16+ Dex [+3]
The earliest you could manage this would be between levels 15 and 16 if you're buying/acquiring them or 13 to 14th if you're having them crafted.
With some creativity in feat selection and finding a way to add various other stats to your AC (Int and Wis are the most obvious), you can decrease your reliance on items somewhat, but that's left as an exercise for the reader.