Not many.
Since Resilient Sphere is essentially a spherical wall of force, and wall of force breaks line of effect, you cannot cast through it. Any spells targeting an enemy through the sphere fizzle.
Summon monster should not be able to summon anything outside of the sphere. The caster needs line of effect to the point where the monster will be summoned, and resilient sphere prevents this.1
Psionics, if they count as spells, also cannot pass through, unless they specifically ignore line of effect (I don't have the occult book so I don't know much about Pathfinder psionics).
Hexes are a gray area. There is very little information on how they interact with magic, other than them being supernatural abilities. Since they should still follow the general magic rules, I believe they must also have line of effect.
Gaze attacks are specifically called out as working through wall of force, but I don't know of a way to gain a gaze attack off the top of my head. There are no spells that qualify as gaze attacks, either, as far as I know.
1 This is debatable, but probably how I would rule it. The question is, is this spell similar at all to teleport or dimension door? It could be considered either. On one hand, you are designating a location, just like Dimension Door. On the other hand, the range is much shorter. The major difference is, Dimension Door targets you, and you are already inside the sphere.
An existing forcecage spell's effect is probably unaffected by an incoming antimagic field spell's effect
A spell that creates multiple walls of force—like the the 4th-level Sor/Wiz spell force chest [evoc] (Spell Compendium 97) or the 7th-level Sor/Wiz spell forcecage [evoc] (PH 233)—requires using the rules for the 5th-level Sor/Wiz spell wall of force [evoc] (PH 298-9), and those include how the wall of force spell's effect interacts with the effect of the 6th-level Sor/Wiz spell antimagic field [abjur] (PH 200), this despite the antimagic field spell's description specifying only the spell wall of force.
The spell forcecage says
Like a wall of force spell, a forcecage resists dispel magic, but it is vulnerable to a disintegrate spell, and it can be destroyed by a sphere of annihilation or a rod of cancellation.
Compare that with the spell wall of force, which says that it's effect
is unaffected by most spells, including dispel magic. However, disintegrate immediately destroys it, as does a rod of cancellation, a sphere of annihilation, or a Mordenkainen’s disjunction spell.
The reader must refer to the spell wall of force to learn more about the spell forcecage (otherwise, for example, the phrase "vulnerable to a disintegrate spell" is meaningless). And, if trying to determine how, for example, the antimagic field spell's effect interacts with the force cage spell's effect, the reader'll end up back at the spell wall of force anyway.
It would take an extremely strict reading of the spell antimagic field—followed by a personal definition of what a forcecage spell's effect's now-mysterious walls of force were actually made of—to have an existing forcecage spell's effect suppressed by an antimagic field spell's effect.
Walls of force usually block line of effect but an antimagic field spell's effect does not
Normally, the antimagic field spell's effect just doesn't care about a magical effect that would block its line of effect: the antimagic field spell's effect causes the existing magical effect to be suppressed. But, like the spell antimagic field says—and the Rules Compendium confirms—, "A wall of force… isn’t affected by antimagic" (11).
Thus even an existing forcecage spell's barred cage effect allows spells to be cast while those within would be, if uncaged, in an antimagic field spell's effect, although usually not beyond the barred cage effect's walls. While an "otherwise solid barrier with a hole of at least 1 square foot through it does not block a spell’s line of effect" (PH 176), the forcecage spell's barred cage effect has only "half-inch wide [bars] with half-inch gaps between them."
However, casting the spell forcecage while within an existing antimagic field spell's effect and picking as the forcecage spell's effect's point of origin a crosshairs within the antimagic field spell's effect means the forcecage spell's effect is suppressed while the antimagic field spell's effect remains. (However, picking for the forcecage spell's effect's point of origin a crosshairs that's outside the antimagic field spell's effect seems to mean the forcecage spell's effect functions normally. The antimagic field spell's effect doesn't block line of effect (RC 11)—an apparent reversal of how the spell antimagic field functioned according to the Dungeons and Dragons, Third Edition FAQ—, but opinions on exactly how this works vary widely.)
Best Answer
Trivial answer: no, because you cannot center antimagic field on a pebble
Antimagic field can only be emanated from the caster him- or herself. You cannot cast it on other creatures or objects, the way you can with, e.g., light. Thus, the sequence you describe cannot take place.
Arcane archers1 and master abjurers2 can stick antimagic field onto other creatures, but not onto objects.
Trivial answer: no, the prismatic field destroys anything you try to bring in
Still, if you were an arcane archer or master abjurer, you could presumably tag some tiny-and-harmless critter with antimagic field (the master abjurer is much nicer about this process) and carry that instead of your pebble.
The violet veil is still going to just destroy it. It won’t make it to you, and the antimagic field will be annihilated along with the kitten (and you become a terrible person, if you were a master abjurer and not already).
Trivial answer: no, the antimagic field is in the prismatic sphere with you
OK, fine: assume an indestructable kitten, or at least one immune to the various effects of prismatic sphere.3
So you have your antimagic field-emanating kitten, you step out of the field (or never entered it, if you were a heartless bast— I mean, arcane archer), and you cast prismatic sphere. You now step towards the kitten, and the antimagic field is held at bay by your prismatic sphere. You step close enough to pick up the kitten,4 at which point your prismatic sphere is wholly enclosed inside an antimagic field. What happens?
The kitten is now inside your prismatic sphere. The prismatic sphere’s edge is a barrier through which the antimagic field cannot penetrate, but our amazing kitten can. Since the source of the antimagic field is now inside the prismatic sphere, it affects you, suppressing your magic, including the prismatic sphere, and freeing the antimagic field to extend to its full radius (rather than being limited by the inside of the prismatic sphere). Your prismatic sphere will return if and only if you step away from the kitten, allowing it to stay outside your prismatic sphere.
Real answer: antimagic field does not block magic, only suppresses it
Ultimately, the thing you were trying to do here is possible. The archmage’s Mastery of Shaping, for instance, could leave a 5-foot hole in the middle of the emanation that is centered on you, leaving you magical, with antimagic field around you. Unfortunately, that’s largely worthless, since magic can pass through an antimagic field. It doesn’t stop magic, just suppresses it while it’s in the antimagic field.
In game terms, antimagic field does not block the line of effect needed to target spells. So you are just as vulnerable to spells cast at you as you were without the antimagic field. Someone standing outside your shell can cast fireball at you, and the little red bead will disappear when it hits the shell – and reappear on your side of the shell, before hitting you in the face.5 Then it blows up, filling your hollow with fire, leaving your antimagic field-filled shell free of any flames, and yet more flames outside the shell.
All it really does is suppress the magic of those standing very close to you – which isn’t awful by any means, but it is a far cry from the invulnerability to spells you might imagine it is.
Also, casting prismatic sphere at this point does an interesting – but not particularly useful – thing. It limits the size of your antimagic field, basically defeating the Widen Spell metamagic you attempted to put on it. Instead of a 20-ft.-radius antimagic field with a 5-ft.-square hole in the middle, you end up with a 10-ft.-radius antimagic field with a 5-ft.-square hole in the middle, and a prismatic sphere around that. Even if you shrunk the prismatic sphere somehow, so it was inside the 5-ft.-square hole, your antimagic field won’t happen outside it.
If you do get a prismatic sphere with an antimagic field outside it, which can be done in more convoluted ways than I feel like getting into, spells can still go right through the antimagic field to attempt to cancel your prismatic sphere. But the prismatic sphere would not “wink out” because of the antimagic field around it, because antimagic field does not block magic, just suppresses it in the specific area covered by the field.6
Arcane archers can shoot an arrow that sticks a spell that ordinarily can only emanate from you, onto a target creature, who emanates it instead of you. The class feature was almost certainly written specifically for antimagic field; there are very few other spells where this would be desirable, but for antimagic field it is very desirable.
A “master abjurer” here is an abjurer who takes the master specialist prestige class in Complete Mage – their major school esoterica class feature allows them to cast abjurations that are normally centered on you (e.g. antimagic field) as a touch-attack spell that emanates from the touched creature. Again, antimagic field was almost-certainly the spell they had in mind when they wrote this.
Arcane archer, that doesn’t make what you did any kind of okay.
For the purposes of this answer, I am ignoring the fact that antimagic field and prismatic sphere have exactly the same size, and assuming the antimagic field is larger; assume Widen Spell was applied to antimagic field if it makes you feel better.
And if you are an arcane archer, you deserve it.
Even if antimagic field did block magic, it’s far from clear to me that the ambient magical energy necessary for spells wouldn’t still exist in the hollow – after all, I can have air inside a container whose walls block air. This is a setting detail that is not addressed by the general rules, and you would have to ask your DM.