What metals aren't dissolved in/attacked by aqua regia?
Keep in mind

The answer will depend upon the reaction conditions. Most importantly,


physical state of the metal: porosity, degree of comminution;
temperature;
mechanical aggravation of metal surface during reaction.


Often times a chemistry text mentions that no reaction occurs. The reaction might still happen. It is just that for the specified parameters the process is meaningless and negligible.



Short overview of aqua regia

Aqua regia is the $3:1$ volumetric mixture of $\ce{HCl}$ and $\ce{HNO3}$. Its additional reactive power draws from monochlorine created in situ.$^{[1]\ [2]}$

$$\ce{HNO3 + 3HCl -&gt; Cl2 + NOCl + H2O\\ NOCl -&gt; Cl + NO}$$




  
  Which metals are impervious to $3:1\ \ce{HCl/HNO3}$?
  


Almost every metal will react with aqua regia provided certain criteria are met.$^{[1]\ [2]}$ The closest you will probably get is ruthenium $\ce{Ru}$, and perhaps osmium $\ce{Os}$. To the best of my knowledge, $\ce{Ru}$ will not react with aqua regia in a meaningful way even if aqua regia is boiling.$^{[2]}$ The difference with $\ce{Os}$ is that powdered osmium is attacked by boiling aqua regia.$^{[1]\ [2]}$

$$\ce{Ru + HNO3 + HCl $\kern.6em\not\kern -.6em \longrightarrow$}$$

$$\ce{\underset{powder}{Os} + $\underbrace{\mathrm{HNO_3}}_{\text{boiling}}$ -&gt; OsO4 + N_xO_y + H2O \\ OsO4 + 2H2O &lt;=&gt; H2[OsO4(OH)2] \\ OsO4 + HCl -&gt;OsO2Cl2 + Cl2 + 2H2O\\ OsO2Cl2 + HCl -&gt;OsCl4 + Cl2 + 2H2O\\ 2OsO2Cl2 + H2O &lt;=&gt; OsO2 + H2[OsO2Cl4] \\ 3OsCl4 + 2H2O &lt;=&gt; OsO2 + 2H2[OsCl6]\\ OsO2 + 6HCl &lt;=&gt; H2[OsCl6] + 2H2O}$$

Brief discussion about the list provided in the comments

Titanium $\ce{Ti}$ does react, and does so at room temperature.

$$\ce{3Ti + $\underbrace{\mathrm{12HCl + 4HNO_3}}_{\text{room temperature}}$ -&gt; 3TiCl4 + 4NO + 8H2O}$$

Rhenium $\ce{Re}$ reacts slowly at room temperature $\ce{-&gt;HReO4}$. This will further react with $\ce{HCl -&gt; ReCl4 + Cl2}$.$^{[2]}$

Hafnium $\ce{Hf}$ does react at room temperature. The reaction is slower than with titanium; overall equation is identical.$^{[2]}$

Tantalum $\ce{Ta}$ reacts when aqua regia is heated to $150\ ^{\circ}\mathrm{C}$. Rhodium $\ce{Rh}$ reacts in a grinded state. As a large compact piece, iridium $\ce{Ir}$ is affected over temperatures of $100\ ^{\circ}\mathrm{C}$. Niobium $\ce{Nb}$ is inert at room temperatures.$^{[2]}$

Summary: ruthenium $\ce{Ru}$ is your best bet.


  
  What makes metals $\ce{Ru}$ and $\ce{Os}$ so stable in aqua regia?
  


The nobility of these metals is not the best explanation. As you correctly pointed out, $\ce{Pt}$ and $\ce{Au}$ react fine. This is direct evidence that for other metals a protective layer should form. The layer varies from metal to metal, but usually is either an oxide (or oxide hydrate), or a chloride.

Effectiveness of mechanical aggravation also points to stable, non-reactive compound formation on the metal's surface.

For ruthenium, as of now I am unsure what this precipitate could be. If anyone has a reference, please edit or leave a comment.$^\text{[reference needed]}$ 


  
  What happens with silver?
  


Silver and aqua regia react very poorly, and for a short amount of time.$^{[2]}$ The culprit is $\ce{AgCl}$ ($K_s = 1.8 \cdot 10^{-10}$)$^{[2]}$. A slow reaction might still take place due to complexation.$^{[2]}$

Surprisingly, silver reacts with $\ce{HBr}$!$^{[2]}$ Its solubility product is even worse, $K_s = 5.0 \cdot 10^{-13}$.$^{[3]}$ My guess is that this layer is not as dense as $\ce{AgCl}$ but this still needs verifying.$^\text{[citation needed]}$



References

(In progress)

$[1]$ N. N. Ahmetov. Anorgaaniline keemia. (1974)

$[2]$ H. Karik, Kalle Truus. Elementide keemia. (2003)

$[3]$ Skoog, West, Holler, Crouch. Fundamentals of Analytical Chemistry. 9th edition. (2014)