Manga Hints
Here is where we look at Palletshipping hints from all the official mangas that have the characters Ash and Gary or some incarnation of them.
Ono's Pikachu series
This is the most popular manga, the one based on the anime. Of course, I first have to adress the fact that in this storyline, Ash supposedly has a crush on May, Gary's sister.
Well, ever heard of a concept called transference?! Ash really loves Gary, but he's in denial, so he transfers his affection for Gary onto his sis, May. And there really is a lot of evidence to back this theory up.
For one thing, every single time he sees her, he asks her how Gary is doing. Really, if every time someone saw you, they asked you about your brother, wouldn't you start to suspect that maybe they liked your brother? Not only that, but at one point Ash has a dream about May, and even in his DREAM the first thing he says to her is, "So, how's Gary?" Then, in the dream, May makes some comment that she will be Ash's sister. Now, unless Ash has a thing for incest, the only way I can think of that May could be his sister is by marriage. As in, if Ash married Gary, she'd be his sister-in-law. I mean really, what kind of guy fantasizes about asking a girl about her brother, and her saying that she will be his sister?! The kind of guy who wants to marry her brother, that's what kind!
Also, whenever Ash thinks of May, he rarely thinks of her by her name. It's always "Gary's sister" this and "Gary's sister" that. Even the picture that Ash carries around of May just happens to have Gary in the backgroud! It really seems that Ono is going out of his way here to make sure that every single aspect of Ash's "crush" on May focuses completely around Gary!

Notice too that Gary is acting pretty jealous!
Ash also seems to think about Gary a lot in other situations. At one point, he thinks of giving up on his quest, but the thought of Gary not respecting him if he did gives him the iniative to keep going. The thought of Gary also inspires him to try to catch a Clefairy.
Another thing is that in this manga, they were definitely freinds origionally. This is proven by an adorable "flashback" scene of them watching the "Super Ball" together, back when they were best friends.

The way Gary is looking at Ash here is adorable!
You've got to wonder why the two would just suddenly stop being friends for no apparent reason. Most Palletshippers support the theory that it was because they had feelings for each other and couldn't handle it.
Later, the two meet up at the Indigo Plateau. Gary actually congratulates Ash on winning his first round, and says he hopes that he and Ash get to battle each other. Ash seems really happy to see Gary and says he'll root for him, and Gary wishes Ash luck!

They're linking arms! CUTE!
A while later Ash is at a research center, which is full of a lot of special stuff that many people never get to see. Ash is supposed to be
listening to Bill tell about some new disoveries, but as soon as he
notices that Gary is there, he gets totally distracted and wanders away from Bill to talk to Gary. They end up having a little competition to see who can catch TR first.
At the very end of the whole seires, Gary and Ash become friends again end up traveling together- alone! The ending is pretty open- the two just head off into the sunset together towards new adventures.

Am I just a hentai, or does what Ash is saying here sound kind of suggestive?
Pokemon Adventures
This manga is based on the video games, and stars Red (based on Ash) and Blue (based on Gary). This is basically an action story and focuses on battling more than on characters, so there aren't a whole lot of hints, but there are a few
pretty good ones. This series is not over yet, so realize that this section will contain spoilers, up through the fourth graphic novel.
The second time that Red and Blue meet up, Red goes to punch Blue, so Blue grabs his hand. However, he holds on a little longer than he really has to, and the two end up just standing there with thier fingers entertwined as they talk.

Their faces are pretty close together, too.
Later, when Red hears that Blue went to the Pokemon tower and didn't come back, he rushes off pretty quickly to try to save him. It turns out Blue has been possessed, and Red saves him.

Red is concerned for Blue.
After Blue is back to normal, they work together to beat Team Rocket. Red thinks to himself that Blue is good, but is too shy to actually say it out loud. After they succeed in defeating TR, Red even tells someone that he didn't really do anything; Blue deserves all the credit.

Is it me, or is Red looking after Blue a little wistfully?
When Red and Blue infiltrate the Team Rocket base together, Red falls down a hole and Blue reaches for him and they call each others' names. Maybe not exactly a hint, but at first Blue WAS determied to do this on his own without Red, and he could have just tried to leave Red for dead inside the compound like Green did.

Well it's cute, anyways.
Later, Blue is literally being suffocated by Koga's Muk, but during this time he's concerned for Red. Koga even says to him, "Do you really think you're in a position to be worrying about your freind?"

Blue thinks of Red.
When Red and Blue meet up at the Pokemon League, they're happy to see each other. They smile and greet each other, and Red winks at Blue!

Blue jumps down to talk to Red.
Before they begin the battle that will determine the champion, they touch hands with each other.

I don't think anyone else did that...
During the battle, Professor Oak goes on a lot about how Red and Blue have taken each others' best traits and incorporated them into themselves...and not just battle strategies either, also thoughts and opinions. A theme here, which was also shown in the story where they infiltrated TR's hideout, seems to be that these two are much better together than they are seperately.

"They know each other like the backs of thier own hands"....^_^
When it's all over, you'd think the loser would be kind of mad and that the winner might be gloating or something, but no, they just kind of look each other and smile and decide to head back home to Pallet- together.

The end of the R/B/G saga.
The next part of the story, the Yellow saga, picks up two years later. Red has gone missing and a person named Yellow has teamed up with Red's Pikachu to help find him. Blue gets word from Professor Oak that Red is missing.

Blue must stop his training to help Red.
Blue drops everything and goes to help look for Red. He meets up with Yellow and the two of them go off together to search for Red.
This manga is currently in the GSC arc, but the first volume of the Yellow arc is as far as I've gotten. When I get more manga, I'll update further.
Pokemon Zensho
This is an offical manga that was never translated. It follows the video games very closely, much more so than Adventures. This manga has a lot of the standard hints, like Satoshi
thinking about Shigeru a lot at random times, and the two fighting and
trying to impress each other.
But the cool thing is that this is another manga where, at the end, the two become freinds again and go off travelling alone together. The way
it happens is, after Satoshi beats Shigeru for the title of Pokemon Master, a pillar starts to fall, and Satoshi dives to knock Shigeru out of the way. At that point they become good freinds and go off together in search of Mewtwo.

Good ending ^_^
Pokemon R/B/Y and G/S/C
Yeah, the video games. Maybe this is just crazy-talk, but I even see a little evidence there. I mean, in RBY, Gary is always waiting around to battle Ash, his teases seem cutesy, and he sometimes even gives helpful advice or compliments. This at least proves that Gary was never meant to be a total enemy to Ash. I also noticed that in GSC, when you run into Gary, the first thing he does is start going on about Ash.
Now, things that happen in the manga and games don't nessesarily have any bearing on the anime, but Ash and Gary having something going on between them seems to be a theme throughout every incarnation of Pokemon in which they appear.