Taking Harry never crossed his mind.
The possibility of taking Harry Horcrux-hunting only occurred to Dumbledore when Harry himself suggested it.
Up until that point Dumbledore was acting solo.
The idea of taking someone else with him simply never occurred to him.
"I think...
perhaps...I may be close to finding another one.
There are hopeful signs.
" "And if you do," said Harry quickly, "can I come with you and help get rid of it?"
Dumbledore looked at Harry very intently for a moment before saying, "Yes, I think so.
" "I can?
" said Harry, thoroughly taken aback.
"Oh yes," said Dumbledore, smiling slightly.
"I think you have earned that right."
Harry felt his heart lift.
It was very good not to hear words of caution and protection for once.
(Half-Blood Prince, Chapter 23, Horcruxes) .
Note that Dumbledore had to think about it when Harry asked him whether he could come the next time.
He looked intently at Harry, judging whether he was ready or not.
That suggests to me that he had never considered taking Harry on a Horcrux mission before that point.
If Harry hadn't asked him then Dumbledore would probably have gone into the cave alone as well.
When Harry did ask him he was happy to take him.
But he needed a prompt for the thought of working with somebody else to even cross his mind.
This is very in-character for Dumbledore.
He's at his most comfortable acting alone.
He also likes keeping his secrets close to his chest.
He only told Harry about the prophecy because he was forced to after it became clear that Voldemort was using Harry's ignorance against him.
He only began teaching Harry about Voldemort and the Horcruxes because he was forced to after he realised he had only a year to live.
There are some supplementary points:
Harry didn't know what a Horcrux was at this point.
Harry would've been thoroughly confused if he'd gone to the Gaunt shack.
What is this place?
What are we doing here?
And so on.
It was far better to explain Voldemort's past to Harry gradually so that Harry understood the significance of the Horcruxes and how to hunt for them rather than just taking him on a mission he didn't understand.
Harry was an emotional wreck who wouldn't have been able to help much.
Harry was cut up over the death of Sirius.
He wasn't in much of a position to help Dumbledore.
Harry was supposed to be at the Dursleys anyway.
Maybe a day-trip wouldn't have broken the Privet Drive charm.
But Dumbledore wouldn't have wanted to take that risk.
If that happened then Dumbledore would have to, like, look after Harry himself.
The Ariana affair shows he would've seen that as a waste of his time.
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Maybe he only decided to bring along a wingman after he semi-failed with Marvolo Gaunt's ring and realized he wasn't invincible?
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Flip your question around.
Dumbledore knew he was seriously powerful.
Remember when talking about the boat he told Harry that it measured magic and with him in it Harry would barely register.
He was also prepared to go toe to toe with V, there is no doubting his ... lack of doubting his ability.
Therefore why would he feel the need to take a 15 year old along with him?
It would just be someone else he would need to look after and handle.
Also, Dumbledore tended to not tell people things unless he had to (remember his brother saying of Dumbledore's expertise in lies and secrecy - he was not an 'open' person).
After getting his finger burned ( snigger ) with the ring he seemed to realise that he had underestimated V's cunning and allowed his emotion to cloud his judgement, therefore he took along someone he trusted whom he could rely on to keep him straight.
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He had seen the ring in the Pensive, so he might have been wondering if it was the Resurrection Stone.
If he was correct, then he would no doubt want to be alone to use it.
Another thing is that, at that time, he had not explained everything to Harry and nor did he need as much help to retrieve the ring.
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I can think of a few good reasons in this case.
He wanted Harry to figure out the stuff about the Horcruxes on his own.
Harry had no idea what Horcruxes were at the time, and Dumbledore wanted to take him through all the backstory before introducing Horcruxes to Harry.
It might be dangerous for Harry.
Who knows what kinds of traps and curses Voldemort might have put on the ring.
This shrewd idea turned out to be a good one, as the ring was in fact cursed.
He doesn't want to introduce Harry to the hallows.
He's trying to save Harry from what happened to him as a child.
Here's a quote to back me up: "
And if you do," said Harry quickly, "can I come with you and help get rid of it?
"
Dumbledore looked at Harry very intently for a moment before saying, "Yes, I think so.
" "I can?
" said Harry, thoroughly taken aback.
"Oh yes," said Dumbledore, smiling slightly.
"I think you have earned that right."
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Leaving aside the plot convenience of Dumbledore being near death for the entire Sixth Year, is there any good reason shown (or inferred from) canon as to why Dumbledore went alone - as opposed to taking Harry with him to retrieve Horcrux #2 (the Gaunts' ring, which turned out to be a Hallow as well) from the Gaunt shack?
Why didn't Dumbledore take Harry with him to the Gaunt shack to retrieve the Ring Horcrux?
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<harry-potter><albus-dumbledore>
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Harry didn't know what a Horcrux was at this point and Dumbledore wanted Harry to figure out the Horcruxes on his own.  It never crosses his mind to take Harry and he wants Harry to work out what the Horcruxes are on his own.
It never crosses his mind to take Harry and he wants Harry to work out what the Horcruxes are on his own.