Travel health insurance in Europe that does not require European Health Insurance Card

TL;DR

• Get the card no matter what if you are travelling for extended periods of time in EU. It doesn't cost anything and helps a great deal if an unfortunate situation occurs.
• If you can't present your card for some reason, you might have to pay ordinary rates (could be very costly!), but you can apply for reimburs*m*nt from your health authorities back home.
• You will not be refused treatment, if your condition is in any way life-threatening (think acute respiratory or debilitating diseases, broken limbs, etc.).

The European Health Insurance Card (EHIC) is a handy piece of plastic that allows you to receive the same medical care in another EU state as you would in your home country. You will receive for free the same services that you would otherwise get free in your home country, however rates for paid services would differ, but will definitely be close to what you would pay home (as opposed to rates given to non-insured individuals). You are also eligible for medical care if you have some chronic condition that requires specialized treatment (for example, kidney dialysis). You'll need to check with your local health authorities about what conditions are covered, in which countries, and whether you'll have to pay for the treatment.

EHIC are free (besides probably a small administrative fee for handling the documentation if you apply offline).

Let's cut to the chase here, and allow me to plug in the European Commission. From their FAQ:

What happens if I travel without the card and need medical treatment?

If the need arises, you will still receive the treatment necessary to enable you to continue your holiday without having to return home for treatment. But remember that the card will make it easier for you to access free medical care on the spot or for you to be reimbursed if you have to pay up front. That is why, when you travel to another Member State of the European Economic Area or Switzerland, whether for private or professional reasons, you are strongly advised to do so with a European Health Insurance Card.

On the topic of reimburs*m*nt, this is where it gets hairy. You are indeed able to ask for reimburs*m*nt from your health insurance agency back home if you present them with the relevant medical and billing documentation. However, this procedure could be quite lengthy (months, I believe), and there's no guarantee that you'll get back everything. Note that it's best that you apply when you are still abroad -- ask the administration in the hospital about details, and consult your local health authorities.

Can a doctor refuse to treat me if I have forgotten my card?

Medical ethics dictate that a doctor cannot refuse to treat you if your state of health requires treatment. However, there is no guarantee that your costs will be reimbursed under the same conditions as if you had been able to prove that you had social security insurance at the time by presenting the card or an equivalent document. The doctor or medical establishment might well ask you to pay the full cost or to pay up front a proportion of the costs which an insured person in that same Member State would not be asked to pay. In an emergency your local health authority might be able to help by faxing or e-mailing you a provisional replacement certificate, which gives you the same protection as the European Health Insurance Card.

and finally,

What should I do if, during or before my visit, I suddenly realise that I have forgotten or lost my card?

If you have forgotten or lost your card, you can ask your local authority to fax or e-mail you a provisional replacement certificate. This will give you the same entitlement as the card does.

(emphasis mine everywhere)

I've lived abroad in another EU country than my own, and I can tell you from experience that in the (fortunately very few) cases I had to use it, it significantly sped up the procedure of receiving medical care. For non life-threatening conditions, you would otherwise wait for some time until they collect and process all necessary information from you. If you have a card, all personal information they need (that is, your names, national ID number, permanent address, etc.) are already in the system.