Although the subreddit asks for extensive responses from historians, I just want to add some remarks as a physicist and mathematician who read extensively about the history of science.

We can safely say that, in math related fields, Gauss and Euler were the most ubiquitous and unparalleled geniuses of their times, with little dispute. Gauss started his career with the discovery of a method to construct the regular 17-gon, a polygon which has 17 equal sides, which was important at a time where euclidean geometry still had a big influence. He proved the fundamental theorem of algebra, had very important contributions in algebra, he made phenomenal contributions to astronomy, discovering Ceres, for example, and meanwhile leaving huge contributions to computational mathematics. His computational ability, speaking of which, with his numeric calculations and photographic memorization of logarithmic tables, was only defied by the invention of an actual computer. Among the most important computational devices invented by him are the method of minimum squares and the normal distribution, which are related. The later was the most important statistical discovery of all time. He invented non euclidean geometry, created the Theorema Egregium and contributed to many other fields, in an extremely vast work.

Euler, who came before him, had the most extensive and impactful mathematical contribution of all history. He worked at so many fields, giving fundamental contributions to each one of them, and creating so many new ones, that it is really hard to talk about it in a just manner. He exploded the reach of analysis with his use of power series, discovering for example the euler number and the exponential, discovering the extremely important relation between exponentials, sines and cosines, and many other extensive works in transcedental functions and differential equations, not to mention the pioneering use of complex numbers. He gave giant contributions to number theory, including ones related with the riemman zeta function, he created graph theory and topology and gave many more contributions to physics.

Newton came before both of them and was kind of their predecessor. In the following centuries after his own, Newton was often considered as the greatest scientific mind that ever lived, specially during the enlightenment. He greatly influenced names such as Spinoza and Voltaire. People seldom realize the crushing extent and geniality of his work, which make him often credited as being the greatest genius who ever lived. He not only discovered the relation between celestial movement and terrestrial gravity, not only made a huge corpus of contributions to fluid physics, not only proved the nature of sound and calculated it's speed, not only discovered the decomposition of light, it's particle properties and a huge corpus of optical phenomena, not only solved the brachistochrone and invented the calculus of variations. The most impressive part is how he did it and the sheer amount of applications and particular, real cases he used to deduce and prove all of it. Newton lived in a time where mathematical language was the language of the Euclid's Elements, a constructive geometrical one, based on the use of the compass and straightedge. The calculus, the most important mathematical discovery since the times of Pythagoras, was Newton's secret weapon, but he did not use it to write his work on physics. This renders an astonishing inventive creativity and genius, to prove things like the effective concentration of the gravitational mass of spherical bodies, using only euclidean geometry. Also, people do not realize the sheer enormity of Newton's works. His main book on mechanics and gravitation, the Principia, has an extremely different way of doing physics than the modern one. It not only expresses a theory based on a minimal amount of evidence. Is has a huge amount of engineering applications and examples which are as distant from the abstract way of doing physics as possible. There are no spherical cows in vacuum. It is a real analysis of hundreds and hundreds of cases, just as Euclid's Elements, in their entirety and complexity, and from the successful application of his principles in these true environments he draws his conclusions.

The most mind boggling thing about Newton, still, is the fact that all of his centuries-impactful work is only a small part of what he produced and devoted his time to. This was only presented to the world in the XX'th century, when the renowned economist Maynard Keynes rediscovered his vast work on alchemy, numerology, Christianity and occultism.

In his time, though, Newton was not so highly regarded and ubiquitous as he came to be, partly because of his extremely timid and secretive character (he almost did not publish the Principia), and because of fierce competition of other names like Leibniz. Not to say that he wasn't recognized, it just took time for his name to become that of the greatest genius who ever lived.

Now, regarding Einstein and Stephen Hawking. You are right that they were often presented popularly as the smartest people alive. I will first compare Einstein to the three names mentioned, and then I talk about Stephen Hawking.

Compared to the work of Newton, Euler or Gauss, Einstein's work is incomparably less robust. It is certainly way smaller, for starters. While Newton wrote huge books, densely filled with his discoveries for hundreds and hundreds of pages, comprising thousands of examples and experiments, Einstein only published some dozens of small papers, in the more modern fashion, and some didactic books targeted at teaching relativity. It is like comparing a mouse to an elephant. The amount of evidence Newton wrote for his mechanics, for example, is one hundred of times more than the amount of evidence put forward for relativity in the entirety of the XXth century. Something little known about Einstein is that, in the sense of volume only, his main contribution is to thermodynamics, with an occasional attention at theoretical physics. Einstein is huge, nonetheless, because of the importance of what he discovered in a dozen of key papers, such as the brownian motion, the photoelectric effect, the thought experiment on the equivalence of mass and internal energy, the thought experiment showing randomness in the Bohr model, and the Bose-Einstein condensate. But his works, erudition and, in my personal opinion, intelligence, are not comparable to the vastness of, say, Henri Poincaré or Karl Jung, who lived at the same time as him, but the sheer surgical impact of his relatively modest work was certainly a good reason for him to gain the press popularity and the influence he achieved.

Now about Stephen Hawking. Although there is in fact such a persona in some segments of pop culture, I don't think he was ever widely popularly recognized in the same way as Newton, Gauss, Euler or Einstein, and for sure he wasn't recognized as such in the scientific community. His appeal to popular fans of star trek back them and big bang theory fans more recently is way more related to the particularities of his condition, his sense of humor, and his literary work. Although he did occupy the lucasian chair of mathematics at oxford, a position once occupied by Newton, it is a position which, again since Newton, was never occupied by someone as ubiquitous as Newton, and, with the exception of George Stokes and Paul Dirac, (and maybe Larmor and Babbage) was not even occupied by very famous physicists. Regarding his work, its highly specialized work in specific fields of theoretical physics, and, with the exception of the Hawking radiation, which is still only a hypothesis and which is kind of a minor phenomenon, it is not even big in those specific fields. In other words, it is kind of something you would need a magnifying glass to sight. His main contribution is to the field of the black wholes, which he helped popularize, more specifically in a subfield which is called black whole thermodynamics, but, even then, Bekenstein is a bigger name. In the purely theoretical branch of physics hawking used to work in, Edward Witten is surely a much, much more impactful and respected name, with a work which is way more vast. But people who contributed to major physics discoveries, like Higgs or Stephen Weinberg, are way higher on the list, just as any other nobel prize winner, mainly in areas such as condensed matter or quantum computation. Back to the theoretical side of things, Roger Penrose is also a bigger name in the same style of scientific divulgation + research that Hawking did. I think Hawking is a phenomenon of the pop culture and media missing having the big genius of an era, just as classical music does not have a modern Beethoven or Mozart, and Hawking fit the stereotype well. But I have to point these things out because having a subculture believe someone to be the smartest human alive, while no serious scientist thinks so, is very different from living in a time, such as Gauss' or Newton's, where hardly anyone could disagree on that person being at least a top candidate. Unlike Newton and Einstein, who made revolutionary contributions that changed physics and science in general forever, Hawking's contribution is of an infinitesimal character to physics in general, just like the contributions of thousands and thousands of other physicists, and it is extremely hard to believe that he will be remembered in one century, except inside a very specific field.