Can you file a patent application yourself, without an attorney, in the US or EU?

In the US, yes. The USPTO allows individual inventors to file pro se, without an attorney, and there is no legal requirement to use one. In the EU, it depends on residence. Applicants domiciled in an EPC contracting state can file and represent themselves before the EPO, but anyone without a residence or principal place of business in an EPC state must appoint a professional representative for all proceedings after the initial filing, so self filing simply isn't legally available to most non European founders. Even where self filing is allowed, claim scope, not paperwork, is where self filed applications most often fall short, since narrow or vague claims are easy to design around.

Sources: USPTO, Pro Se Assistance Program, EPC Article 133, General principles of representation.