Can I have extra minecraft inventory?

There is no way in vanilla Minecraft to expand your inventory, whether by commands or by other means.

If you wish to have a larger inventory, you could try making an Ender Chest and using one slot of your inventory to carry it around. Note that you need a silk touch pick in order to pick it back up.

This would mean you can have up to 61 inventory slots (63 minus two slots to carry the Ender Chest and pick) and 4 armour slots total. This is 25 more than the normal 36 slots.

Shulker Boxes deserve a dedicated answer of their own.

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It is now possible to have extra inventory slots (as of 1.11).

You simply need a Shulker Box...

• 27 item slots
• can be placed and broken
• they keep their inventory when broken, so you can access it after placing the box again
• can be placed in your inventory and other containers
• can not be placed in other Shulker Boxes
• unlike chests the Shulker Box will be facing away from the block it's placed against
• you can give it a name with a Name Tag and Anvil

I don't think that there is a way to have more room in your own inventory.
Instead, there are other ways to carry more:

1. Build a "donkey train". Attach chests to some donkeys (or mules), mount one of them, then use a lead to pull the others with you. With this method, you can significantly increase your storage slots, with 15 per donkey.
2. Use Ender Chests. They act like "cloud storage", meaning that the items you put in one, can also be picked up from another one that's far, far away. Furthermore, what you store in them is private, giving you a world-wide storage available only for you, increasing your storage slots by 27.
3. Use Shulker Boxes. They condense a whole chest's worth of storage into one block that cannot be stacked, but can be placed down and picked back up with any tool without dropping their contents.

4. Carry your items in "packed" states, like:

   • Blocks of ore instead of materials (ores)
   • Bones instead of bone meal
   • Logs instead of planks
   • Etc.

   This method may help you to preserve your already existing slots. Sometimes it may help you to make massive slot savings, or sometimes it may be insignificant.