The size of the header will depend on what load it is supporting (roof, floor, etc. and the clear span of that framing), what building code applies, what snow loading is appropriate, and the roof slope.

You should also tell us if there can be a stud or two between the windows since that will greatly reduce the header size.

It is sometimes possible to put the header above the top plate if floor joists run parallel to the wall.

It will also be of interest, what openings already exist across this wall and where they are. Start from either corner.

Include the proposed window height.

Also, what are the interior and exterior wall finish material.

I assume that the double plate counts for 3" of the total hight of the beam?

This assumption is generally wrong, because the fastening between the two pieces of the top plate, and between the top plate and the header, are not likely to be nearly so perfect as to make them act as a single piece. A human (properly trained, of course) can karate-chop through a stack of thin boards, even though he would break his arm hitting the same stack if the layers were glued together.

One old rule of thumb I read somewhere was that doubled 2x headers should be an inch high for every foot in width, e.g. you would need an 8" header for an 8' opening. It's imprecise, but it would work.

If you can't get the dimensions you want in common dimensional wood, LVLs and steel are possibilities.

The rule of thumb mentioned is actually only for beams limited by deflection and it is the depth of the beam in inches = 1.1 X the span in ft. That would mean an 8-0 beam would need to be a minimum of 8.8 inches deep and that means a nominal 10 inch depth assuming the beam already meets or exceeds the bending moment requirements. Unfortunately, few headers are limited by deflection because of their relatively short lengths so this rule of thumb isn't very useful.

There are other rules of thumb for sizing floor joists and rafters since they have relatively predictable loadings but there can be none for beams and headers because their sizes are entirely dependent on the different loading conditions of different building heights, widths and snow loads. A rule of thumb can't accommodate a 300% difference in loading.

A small one story house with no snow loading might be able to use a double 2x8 header for an 8-0 opening but a 2-story house in a snow zone might need double 2x10's and some would need double 2x12's.

LVL's have about double the bending strength of dimension lumber and about 50% greater stiffness but they require an engineer to size them since they are not included in building code framing tables.

A steel plate can be placed between the members (betterheader.com) but they are expensive.

Putting a post between two 3-8 windows is the easiest way to reduce the size of the header.

You can use the 'per se' headers in the building code.

Things like flitch beams (steel plates between wood members) normally require an enineers's stamp to be approved since there are many pitfalls in their design.

Things like hole tolerances, fastener diameters, loading of the wood by buckling of the steel, etc.
Even among engineers there is not uniform agreement about some items.

The building code has a table for sizing headers but it is useless unless you know the ground snow load, the width of the building and the floors and ceilings supported in addition to the roof.

Use of these tables sometimes results in slightly oversized headers and they do not include LVL's or other manufactured beams. To minimize the depth of a header it is necessary to design it using "accepted engineering practice" which means using the standard formulas for sizing beams for bending, deflection, and shear. In almost every state I have practiced the local building department requires an engineer 's stamp as evidence of competence to do this.

"jon1270" & "brickeyee"

The "rule of thumb" I.E. the 'per se' method are commonly used here on the truly left coast.
The 1" in height for each lineal foot of opening is allowed to be met by either "headers" of #1 or #2 and better grade and/or 1/2" plywood or OSB between 2, 2x dimensioned lumber fastened with opposing rows of min. 10D nails.
Headers in excess of 6' require double trimmers/jack studs at each end.

A civil/structural engineers stamp is seldom required on single story dwelling except in unusual circumstances, such
as large open areas, and "shotgun" homes where the length exceeds the width by more than 3 times without a shear wall.

To give you an idea of how ineffective structural "rules of thumb" are, for a one story house 20 ft wide you would need to multiply the 8 ft opening by 1.25 to get a header size that meets the IRC header tables. For a house 28 ft wide you would need to use a multiple of 1.5

"The 1" in height for each lineal foot of opening is allowed to be met by either "headers" of #1 or #2 and better grade and/or 1/2" plywood or OSB between 2, 2x dimensioned lumber fastened with opposing rows of min. 10D nails.
Headers in excess of 6' require double trimmers/jack studs at each end."

I am well aware of the the 'guidelines' form the code (the 'per se' rules).
As a PE, stamp in hand, I preform the calculations all the time.
Some are fine (or even over specified by the code) but a decent number are undersized.
If there is no reason to decrease an over sized header I unusually just let them go. One less 'special issue' to deal with.
If the code version is undersized I detail it out and ALWAYS use the larger header.
Very few inspectors even notice, but there is one guy who asked years ago why I used the larger size.
After an explanation he even asked for a calculation sheet.

He came back latter, shook his head, and noted he could not fail other houses since they met the letter of the code. He agreed there was an issue.

"I am well aware of the the 'guidelines' form the code (the 'per se' rules)."

I know, and should the broader audience find cause to ponder, inquire and discuss possibilities instead of arbitrarily arriving at a conclusion, the effort will have been served.

The OP may even decide to rejoin, eh?