How much arbitrary data I can embed in a transaction?
Does Monero have something something similar to Bitcoin's OP_RETURN?
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Sign up to join this communityAn arbitrary amount of arbitrary data can be added in a transaction. The extra field contains data with a high level structure. Several fields are predefined, such as public key, or extra nonce. In order to add custom data there, you would need to define another data type, and store it in that extra field. Minergate recently did just this, adding 32 bytes of data with unknown semantics.
Note that since Monero has transactions fees which depend on transaction size, the more data you add to the extra field, the more you will have to pay in fees (currently 0.01 monero per kB).