The Beguiler class is a fixed-list spontaneous arcane caster; a Beguiler PC gets access to spells from the Beguiler list as it levels up and can cast them spontaneously.
A beguiler casts arcane spells, which are drawn from the beguiler spell list. When you gain access to a new level of spells, you automatically know all the spells for that level on the beguiler's spell list. You can cast any spell you know without preparing it ahead of time. Essentially, your spell list is the same as your spells known list.
A beguiler need not prepare spells in advance. You can cast any spell you know at any time, assuming you have not yet used up your spells per day for that spell level. A beguiler's caster level is equivalent to his class level.
They get a wide and varied list which is, unfortunately, pretty hard to expand. The Collegiate Wizard feat is pretty specific in only helping Wizards get more spells. But another feat looks interesting.
The Greyhawk Method feat isn't nearly as discriminatory as Collegiate Wizard:
Prerequisite: Ability to prepare and cast arcane spells.
Benefit: You gain four new spells of your choice to add to your spellbook each time you gain a level in a class that allows you to prepare and cast arcane spells (such as wizard). These spells represent the results of your research, and must be of spell levels you can cast. If you have chosen to specialize in a school of magic, two of the four free spells must be from your specialty school. [...]
Unfortunately, it requires the ability to prepare arcane spells, which Beguilers cannot. However...
The Arcane Preparation feat seems to help with that:
Prerequisite: Ability to cast arcane spells without preparation.
Benefit: Each day, you can use one or more of your spell slots to prepare spells you know, usually for the purpose of applying a metamagic feat to the spell-but without an increase in its casting time. Preparing a spell uses a spell slot of the appropriate level, and once prepared, that slot can't be used for anything else until the prepared spell is cast.
This seems to allow a Beguiler to take Greyhawk Method, as the beguiler now has the "Ability to prepare and cast arcane spells".
What would taking this accomplish, if anything?
The Beguiler has no spellbook to add spells to, nor does (for instance) a Wizard with the Eidetic Spellcaster ACF, in spite of both being classes allowing the preparation and casting of arcane spells, at least once the Beguiler has gotten Arcane Preparation.
Can a Beguiler benefit from the Greyhawk Method?
And if so, which spells would the Beguiler be able to learn, since their own list is already fully known?