Elven Wizard Generalist number of new spells per level?

Question

The Elven Wizard 1st level Racial Substitution 'Generalist Wizardy' (Races of the Wild) says:

At each new wizard level, she gains one extra spell of any spell level that she can cast.

which one of my players has found ambiguous. It could either mean: 1. one extra spell of [every] spell level castable or 2. one extra spell of any [one] spell level castable.

How many new spells should an Elven Generalist Wizard gain at first, second and third character levels?

Answer 1

The "any" is used to denote a choice, rather than inclusion. That is, your second reading is correct - it's any one level.

This is not necessarily clear from the sentence in the Elven Generalist Wizard description on its own, but it does become so when viewed in context with the rest of the system. Notice, for example, that this is the same language used by the Wizard class itself:

At each new wizard level, she gains two new spells of any spell level or levels that she can cast (based on her new wizard level) for her spellbook.

Here, the "spell level or levels" part makes it clear that the total number of spells gained is two. (Side note: This is generally not contested, see for example that the entirety of this question takes it for granted.) The wording in Races of the Wild is the same, only the lack of pluralization hides the context. Nevertheless, the context exists, and can lead us to the correct reading.

It also happens that this is the far more balanced result.

Answer 2

So, the text in the RotW is, well, this:

    Generalist Wizardry: A 1st-level elf wizard begins play
with one extra 1st-level spell in her spellbook. At each new
wizard level, she gains one extra spell of any spell level that
she can cast. This represents the additional elven insight and
experience with arcane magic.
The elf wizard may also prepare one additional spell of
her highest spell level each day. Unlike the specialist wizard
ability, this spell may be of any school.

In other words, we've got two parts here: bonus spells in the spellbook and bonus spells per day

Considering the first part:

A 1st-level elf wizard begins play with one extra 1st-level spell in her spellbook. At each new wizard level, she gains one extra spell of any spell level that she can cast

In other words, you've got one extra spell in a spellbook on every class level. It narrows the choice to the spells the character is able to cast.

The second part:

The elf wizard may also prepare one additional spell of her highest spell level each day. Unlike the specialist wizard ability, this spell may be of any school.

This one pretty much narrows it as well - you have only one extra slot, in which you may memorize the spell of your max level, or lower. Just one

P.S. Here's a tip. By taking the Precocious Apprentice (Complete Arcane, page 181), you get one spell and one spell slot of the second level. That is great by itself, and it greatly stacks with Elven Wizard Generalist, since by taking it you can cast the 2nd level spells from level one. As well as learning them, even if your DM says that Precocious Apprentice does not allow you to learn those spells by generic wizardy means:

A 1st-level elf wizard begins play with one extra 1st-level spell in her spellbook. At each new wizard level, she gains one extra spell of any spell level that she can cast

As well as gaining another spell slot for them. Of course, on 1st-2nd levels you'll have to make a caster check to cast them at all, but by the third character level, even if you count only those spells that you get by Generalist Wizardy, you have 3 extra spells and two extra slots for 2nd level spells (one of the slots stays there for all the further play, and the other gets higher and higher, untill you get the 9th spell level). Pretty neat And yeah, just one spell by day is not as great as "1 extra spell on every spell level", but at least you can memorize the previous level spells in it, if you truly need to 8)