How does enchantment combining with an anvil work?

You need 2 enchantments of the same level to get an enchantment of the next level up. There is a cap to the max level enchant you can have. Every time you combine/repair an item in the anvil, the experience cost goes up, and when that goes past 39 levels, you cannot combine with that item again.You can only combine 2 of the same items to combine enchants, except for enchanted books.

When combining enchantments, the enchantment caps out at a certain number. These are as follows:

Armor:

• Protection IV
• Fire Protection IV
• Explosion/Blast Protection IV
• Projectile Protection IV
• Feather Falling IV
• Respiration III
• Aqua Affinity 1
• Thorns III
• Depth Strider III

Swords/Axes:

• Sharpness V
• Smite V
• Bane of Arthropods V
• Knockback II
• Fire Aspect II
• Looting III

Bow:

• Power V
• Punch II
• Flame I
• Infinity I

Rod:

• Luck of the sea III
• Lure III

Other/Most/All:

• Unbreaking III
• Efficiency V
• Silk Touch I
• Fortune III

Enchantments of the same level can be combined to get an enchantment 1 level higher; combining enchantments of different levels always yields an item with the higher of the two levels, giving no increase.

Trying to combine any enchantment that would give you past the numbers above will just give you the cap, EX: Unbreaking III and another Unbreaking III would only give you Unbreaking III, instead of Unbreaking IV

You cannot put Silk Touch and Fortune on the same tool, you cannot put more then 1 different type of Protection on the same piece of Armor, and you can not have any combination of Sharpness, Bane of Arthropods, or Smite on a weapon. All these enchantments are standalone and may override another if put on an item.

Enchanted books can be used at an anvil to combine with another item. Enchantements will apply to the right objects (for example,you can't apply Fortune on boots,except on creative mode) or combine with another book. Note that if a book has multiple enchantments,and just some of them can be applied to an object,the others will be lost,so choose carefully which enchants to get out of the book. Be careful,though,the book enchantment method is level-consuming,and the anvil has a limit of 39 levels at once! In case you ever go over the limit,you will better do the work in more sessions. Anyway,if you combine 2 equal enchantments,say,Protection 3,you will get Protection 4,not Protection 6. You will always get the next level if you do combine 2 enchantments of a level. There are also some limits provided in the list below:

• Protection 4
• Fortune 3
• Unbreaking 3
• Efficiency 5
• Sharpness 5
• Power 5
• Feather Falling 4
• Respiration 3

Notes: These are only some of the many enchantments,see more on the wiki. As you stated,some enchantments do not stack,such as Sharpness with Smite. Also,combining enchantments of different levels will result in receiving an item with the biggest enchant of those two. (Example: Fortune 1 + Fortune 2 = Fortune 2) Also,you can oly combine the enchants of the item of the same type or an item with an enchanted book.