Are there proteins with permanently bound magnesium?
From the look of things you may actually be referring to a lot of different proteins, however I managed to
grab two that I found fascinating:


Enolase


A glycolytic enzyme which catalyzes the reversible dehydration of 2-phosphoglycerate to phosphoenolpyruvate:

 

Credits: Lehninger Principles of Biochemistry page 213 ("Enzymes")


In the enzyme active site, 2-phosphoglycerate undergoes strong ionic interactions with two bound Mg2+ ions causing the resulting intermediate to be easily abstracted.





GTPase specifically  ARF1 (ARF for ADP-ribosylation factor). These are important in priming phase in the formation of Clathrin-coated vesicles (CCVs).


Clathrin itself is  a protein that forms a polyhedral framework around vesicles that transport TM, GPIlinked, and secreted proteins from the Golgi to the plasma
membrane.



X-ray structures of ARF1.GDP and ARF1.GTP, were determined by Dagmar Ringe and by Jonathan Goldberg:



Credits Voet and Voet Biochemistry 4th ed page 435 ("Membrane Assembly and Protein Targeting")

X-ray structures of (a) ARF1.GDP (left) and (b)ARF1.GMPPNP (right) (GMPPNP is a nonhydrolyzable GTP analog in which the O atom linking GTP’s beta- and gamma-phosphorus atoms is replaced by an NH group.) 

The bound nucleotides are drawn in stick form in white with their phosphorus atoms magenta and their bound $\ce{Mg^2+}$ ions shown as lavender spheres.

In ARF1.GDP, the protein’s N-terminal helix (red) together with its
covalently linked myristoyl group (not present in the X-ray structures) are bound in a shallow hydrophobic groove on the
surface of the protein formed in part by the residues of loop gamma 3.