[RPG] Is it possible to increase the number of spell slots a wizard has

Yes, but only 1st-level spells

PurpleVermont has already argued why such a multiclass could only write down spells of a level they could cast as a Wizard (so actually up to 2nd level, if they take Wizard 3). I disagree with Vermont's argument that stolen spells could not be transcribed at all, however.

The rules don't clearly state that you have to have seen a spell written down to be able to transcribe it, that's a matter of interpretation--and I disagree with the interpretation:

Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

The fact that different wizards use different notation implies that the written form of the spell is not an inherent part of the magic, merely an aide-memoire. And the requirement to practice the spell until you understand its V/S requirements -- precisely the thing which the Arcane Trickster can instantly intuit -- strongly suggests that this is the most important part of "learning" a spell, and the rest is merely paperwork that your first-level Wizard training would absolutely prepare you for.

There are, but not much.

These are the two items I could remember:

• Pearls of Power (DMG): These items let you reprepare a spent slot with the magic previously spent. It doesn't add more slots, but adds versatility so you can prepare less repeated spells and use the same spells slots more than once. Best of all, doesn't occupy a magic item slot.
• Ring of Wizardry (DMG): The dream of every wizard apprentice that unfortunately only some masters end up grabbing one. Each of those rings are associated to a spell level, at that level, your base slots are doubled, not extra slots from high intelligence or school specialization, just the ones from the Wizard table.

Other items may improve your spell management during the day without increasing your slots directly:

• Wands: If you use a spell frequently and if caster level or save are not important for that spell (ex: Snake Swiftness, Celerity etc), using a wand extend greatly your spell management and add versatility for you as it opens slots that you would prepare with them.
• Staves: Same as wands but better since they use your caster level if higher and your spell save DC. If not enough, staves can have multiple similarly thematic spells on them.

All other slot increase methods I could remember came from feats (ex: Extra Slot), class choices (ex: School Specialization), prestige classes (ex: Ultimate Magus) or atribute increase (ex: Headband of Intellect).