|Style over substance|
The Expanding Earth hypothesis, first proposed by Robert Mantovani in 1889, attempts to explain continental drift, but without the actual continental drift.
This hypothesis was actually considered not unreasonable before the acceptance of plate tectonics, so this could then have been considered to be a protoscience. These days, it's cranks all the way down.
 The idea
The hypothesis posits that the Earth is growing as a result of the expansion of the seafloor. According to its proponents, an "expanding Earth" model explains why the outlines of continents that were separated by continental drift do not quite match up (most people cite erosion as the major cause of this). If the Earth were, say, 80% smaller, it is claimed that South America and Africa would fit together like pieces in a jigsaw puzzle.
There are several versions of the hypothesis, from one that claims that the planet's mass has remained constant (and surface gravity has therefore decreased), to another that holds that the planet's mass has increased (where the extra mass came from is completely unaccounted for), to yet another that holds that the gravitational constant of the universe has changed.
 Problems with the hypothesis
As mentioned above, the idea of an expanding earth has several problems and very little evidence to back it up. For instance, in continental drift theory, we have measurements of the rate of plate movement (done by fancy satellites and lasers) that show the speed of movement. This movement, projected back, gives us the last time the continents were together. That these figures match up with geological strata and the fossil record is another big hint that we have this right.
Although some observations that lead to continental drift can also - by a stretch of the imagination - be explained by an expanding earth, no positive evidence exists that the earth is expanding. The issue as to whether the earth has increased in mass (and where it came from) if gravity changed (and how) and if the Earth's density is altering are still issues that the expanding-earth idea fails to address. How the concept fits in with the development of the solar system is also unknown, as presumably other planets and moons would also undergo expansion if it was possible.
The "theory" also utterly fails to explain places where plates are crashing into each other - most notably, the creation of the Alps and the Himalayas due to the African and Indian plates, respectively, moving north and encountering the (relatively) immovable object of Eurasia.
 Present-day advocates
The most prominent current advocate is superhero comics artist Neal Adams, who is otherwise a sound enough fellow and does great comics but has just this little bit of batshittery. He posits new mass being created by electron-positron pair production in the core of the Earth.
There is apparently a group called the Growing Earth Consortium.