How many bookshelves for level-30 enchanting?

The rule

Each bookshelf within 1 block of your enchanting table increments the max enchant level by 2 (capped at 30). You need 15 bookshelves to hit the cap; one extra bookshelf does nothing, but you may add up to 30 if you like the aesthetic.

Bookshelves must be placed in the ring 1 block away from the table - either 1 block higher (to make a 2-tall wall) or at table-level - and the air gap between the shelf and the table must not be obstructed by any non-transparent block.

Standard 15-shelf layout

. B B B B B .
B . . . . . B
B . . T . . B
B . . . . . B
. B B B B B .

T = enchanting table
B = bookshelf (1 block from T, optionally 2 blocks tall)
. = air or torch
      

That's 14 bookshelves at floor level. Add 1 more (anywhere in the ring, stacked 2 high) to hit 15. To make 30 enchants reliable, also fully double the ring's height. The level itself doesn't change after 15 shelves, but more shelves give the algorithm more options to surface the rare enchantments you actually want.

What level-30 actually buys you

Level-30 enchants roll the strongest single enchantments in the game, but you never get the full max-loadout from one click - you'll always layer additional books on the anvil. That's where the calculator earns its keep: it tells you how cheaply you can combine those books.