Probably more integration than we have now, but not full integration like the US. I think something one might call a confederation is likely. Unified military, more integrated economies (stuff like having a common framework for company registrations, not tax but taxation systems), shared system for identification/passports, maybe an EU-tax (similar to how most countries have something like municipality/city tax, regional tax and state tax, some of that could be moved to an EU-wide pool of money).
But I don't think we'll see taxation and budgets left over completely to the EU, education, healthcare etc. Just unify the boring stuff and make integration better.