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In about 5 billion years the Sun is predicted to become a red giant and have more than 200 times its current size, reaching a radius of about 5 AU when largest.
I wonder what spectral class the Sun would have then, I think somewhere from M1III to M5III. Do we know a red giant star that is similar to what the Sun is predicted to become?
Models for the future behaviour of the Sun do vary, mainly as a result of uncertainty of mass loss during the red giant (H shell burning) and asymptotic red giant (H+He shell burning) phases.
A highly cited paper by Schroeder & Smith 2008 claims that the Sun will reach its maximum size of about $256 R_{\odot}$ (1.18 au) at the very tip of the red giant branch (and not at the end of the asymptotic giant branch phase, which is suggested by some other models). This maximum size will occur 7.6 billion years in the future (not the 5 billion years of popular literature), when the Sun's mass will be reduced to about $0.7M_{\odot}$ and have a surface temperature of 2600 K (or about 2300 Celsius). This would have a spectral type of M6III, or perhaps even M6I.
Do we know of a star similar to this? It depends what you mean by similar, but there are unlikely to be any stars like this in our Galaxy. The reason for this is that star formation began in our Galaxy about 12 billion years ago. But a 1 solar mass star like the Sun requires about 12 billion years (or a bit more) to reach the tip of the red giant branch. Even were this just about possible time-wise and we were to find a red (super)giant star of about $0.7 M_{\odot}$, it would be highly unlikely that the star would have a similar chemical composition to the Sun. That is because stars born early in the life of our Galaxy would have very low concentrations of metals like iron and nickel that are only produced inside stars and only present in the interstellar medium when a generation of stars have lived and died.
So my perhaps pedantic answer, is that there aren't any big red giants we can see now that started life as 1 solar-mass stars and have a similar chemical composition to the Sun.
Arcturus is a RGB star, probably fairly similar how the Sun will look when it becomes a red giant. Arcturus is slightly more massive than the Sun ($m_{\rm Arc}=1.08 m_{\odot}$), but the main difference is the lower metallicity of ${\rm [Fe/H]}\approx-0.5$.
This low metallicity reduces the opacity in the stellar radiative zone (which fills a significant portion of its volume, even in the RGB phase). The reduced opacity increases the efficiency of radiative energy escape from the core, which makes Arcturus probably a bit smaller and more luminous than what the Sun will become. So to me it seems like the Sun should become an early type K-star for most of its RGB lifetime. Stars that become M-type giants would be more massive ones, like Betelgeuse.
The drastic expansion of a red giant however appears once it enters its relatively short AGB phase, see also here. This is the final phase of its life, accompanied by nebula-producing significant mass loss. As you see, the solar temperature will remain in the K-type range during that phase, before it moves bluewards when shedding the last gaseous layers, and finally exposing the white dwarf at its center.
