Your question is not so clear. The equation degree can be as big as you want. Le's take as an example the 3D space:

• we have a plane surface, Ax + By + Cz = D, which is a 1st degree eq.,
• or a spherical surface, x2 + y2 + z2 = D, 2nd degree eq.,
• but nothing interdicts higher degree equations, e.g.  Axk + Bym + Czn= D, where k, m, n, are arbitrary numbers, positive, negative, integer or fractional.

 I am looking for the biggest implicit eq. of the surface where it has integer degree.

And surface should be minimal.

Ah, you want a minimal surface? But, minimal with respect to what? A geometrical point has surface zero. A line in the 3D space has surface zero.

Also, which type of surface, open, closed? If the surface is open, then is it limited by a certain contour? If the surface is closed, then what about the volume it surrounds?

So, please give more indications.

I want  a surface zero mean curvature (H=0) surface has the biggest implicit eq. degree.. 

Open or closed is not problem.