How to embed custom HTML code with no plugins?

Sounds like you are using the Visual side of the editor.

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You need to use the Text side.

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Get it?

Update:

Preview won’t show embedded objects. Publish it privately if you absolutely must see it before anyone else, and need to see embedded bits like <object> tags.

Update #2:

Preview will show many embedded objects now as of WP 4.0. And many embeds can occur from merely a URL (tweets, youtube, for example).