Are there weapons in Pathfinder that have reach and that can also be used to make attacks against adjacent targets, like the D&D 3.5 spiked chain can? I'm looking for actual weapons that do this; I don't want to rely on feats or spells.
No - there is no single weapon in Pathfinder for a Medium sized creature that naturally threatens a 10ft area.
Any reach weapon plus spiked armour or a one-handed reach weapon plus spiked gauntlet (SRD Weapons Link) will do this fine however.
This doesn't require Two-Weapon-Fighting as long as you don't plan on attacking with both weapons in the same round. Although the spiked gauntlet/armour aren't optimal damage-wise they do let you threaten and attack adjacent squares.
As noted in another answer, whips fit the brief of attacking anywhere within a 15ft area, but they do not threaten, have poor damage and provoke attacks of opportunity when used. With significant feat investment they can be an interesting choice, but it doesn't sound like they meet your requirements.
The whip can reach any target in its 15-foot range. But:
it's an exotic weapon (so you need Exotic Weapon Proficiency (Whip) to avoid the -4);
it deals 1d3 non-lethal damage, and only if the target is not armored (a armor bonus of +1 or a natural armor of +3 is enough to be immune), unless you get Whip Mastery or a Scorpion Whip (1d4 lethal, but, as I understand, needs its own Exotic Weapon Proficiency);
it doesn't threaten, unless you spend a feat on Improved Whip Mastery (but only at natural reach + 5-foot, so 10-foot for a medium size).
So, quite a feat burner to make it efficient, but it exist.
No, but why do you care? I've seen people express a lot of concern about not being able to threaten adjacent while wielding a polearm. Many players hesitate to use a reach weapon because of this issue. In my experience this concern is not justified. In practice this issue rarely comes up in actual Pathfinder combat. My first few polearm-wielding Pathfinder PCs always had a way to threaten adjacent (e.g. armor spikes, improved unarmed attack, etc). Eventually I stopped bothering, because it hardly ever came up and didn't matter on those rare occasions. After about 25 levels of PFS play I decided that the extra weight or hassle of a 'threaten adjacent' weapon was not worth it.
@YogoZuno: I did not say that these situations never arose, only that they are such uncommon edge cases, and rarely important ones, that I've learned one can safely ignore the issue. E.g. My 3rd level Cleric just inflicted 47 hp damage with two AoOs, so I'm OK not getting off that adjacent 3rd AoO with an inferior low-damage weapon. I find reach PCs so effective there is no need to squeeze out that tiny bit of uncommon extra damage from adjacent AoOs.