Life is a state, an experience that everyone has and thinks he can recognize in other people and things. We think life is very important, yet no one can define it. So, assuming that its origin is a fully natural event is a leap of faith, like assuming that consciousness is a fully natural phenomenon.
Disconcertingly, there is no built-in expiration date for this position. There is no point at which, all natural options having been exhausted, we are free to reconsider it, even if we fail to find a naturalistic answer indefinitely.
Some theorists hold that life is produced by a yet undiscovered law of nature. But, unlike other laws of nature, where we have some idea how they work even before we have a succinct statement of the law, we have no idea what this law could be. So, not surprisingly, the law-approach — while immensely attractive in principle — remains a minority choice because it offers so little direction.
What about a chance origin of life? In this view, life is an unrepeatable confluence of accidental events. We are told that theories in science should be beautiful, and chance does offer a sort of bleak, existential grandeur. As Jacques Monod (1910-1976), who won the Nobel Prize in 1965, explains:

The universe was not pregnant with life nor the biosphere with man. Our number came up in the Monte Carlo game.

He concluded his book Chance and Necessity with this observation:

The ancient covenant is in pieces; man at last knows that he is alone in the unfeeling immensity of the universe, out of which he emerged only by chance. Neither his destiny nor his duty have been written down. The kingdom above or the darkness below: it is for him to choose.

But a chance origin also offers a practical advantage. Precisely because life is so complex, a great many ideas can be researched. And the field is still at the starting gate. When Harvard chemist George Whitesides received the coveted Priestley medal in 2007, he said, “Most chemists believe, as do I, that life emerged spontaneously from mixtures of molecules in the prebiotic Earth. How? I have no idea.” He did know what he thought of the cell, however:

The cell is a bag — a bag containing smaller bags and helpfully organizing spaghetti — filled with a Jell-O of reacting chemicals and somehow able to replicate itself.

The secret is in that “helpfully organizing spaghetti.”
Although chance-based proposals are far more numerous, they face the same key hurdles as law-based ones, including:
Life started quickly. Some recent research has identified life on land over two billion years ago, consisting of a fungus whose central cavity was filled with symbiotic bacteria. Some Australian fossils are said to date back to 3.5 billion years old, not long after the cooling of Earth’s crust from the bombardment by planetesimals (rocky objects) at 3.85 billion years ago. Of course, some such current findings may be revised. But the general trend has been to the discovery of ever-earlier instances of life on Earth. That means that very complex and precise sequences of events must have taken place in a short period of time. Also, we don’t know in detail what the conditions on early Earth were like so it is difficult to refine the search by ruling out whole classes of theories.
Replication, by itself, tends toward greater simplicity rather than greater complexity. As Brian C. Goodwin recounts:

In a classic experiment, Spiegelman in 1967 showed what happens to a molecular replicating system in a test tube, without any cellular organization around it. … these initial templates did not stay the same; they were not accurately copied. They got shorter and shorter until they reached the minimal size compatible with the sequence retaining self-copying properties. And as they got shorter, the copying process went faster.¹

Living bodies have intricate machinery that prevents this dreadful simplification. But how did such machinery form, embodying a reversal of the drive to simplicity?
Life as we know it is transmitted by language-like codes. This is the most serious problem. Origin-of-life researcher Carl Woese (1928-2012) has called it the “dark side of molecular biology.” As mathematician and philosopher David Berlinski puts the difficulty:

One half of the modern system of coded chemistry — the genetic code and the sequences it conveys — is, from a chemical perspective, arbitrary. The other half of the system of coded chemistry — the activity of the proteins — is, from a chemical perspective, necessary. In life, the two halves are coordinated. The problem follows: how did that — the whole system — get here?

We are told that it is a language but that no one invented it. That it has no meaning, but it utters living beings. Under the circumstances, even the most plausible theories of life’s origin are a leap of faith that inanimate nature can do what seems impossible to intelligence.
Despite all this, some of the theories enjoy considerable prestige.
References:
(1) Brian Goodwin, How the Leopard Changed Its Spots: The Evolution of Complexity (New York: Scribner’s, 1994), 35-36. Quoted in Bruce L. Gordon and William A. Dembski, The Nature of Nature: Examining the Role of Naturalism in Science (Wilmington, DE: ISI Books, 2011), p. 388.
Editor’s note: Here are links to the whole “Science Fictions Origin of Life” series.