Alien life with different chemistry from ourselves is a very interesting topic*, but consider: life based on different states of matter.
Organisms on Earth are built out of a mixture of liquids and solids, with chemical interactions occurring in a liquid medium. On a smaller scale, we are made out of cells, which exploit the immiscibility of lipids and water to enclose a watery salty medium within lipid membranes. On a large scale, this results in mostly solid but squishy bodies. This is a good compromise to move chemicals around while also allowing enough concentration to have them react.
Can we accomplish that in a different way?
Very low temperature life kept together by weaker atomic bonds. The lasting parts of our bodies are kept together by rather strong kinds of bonds: covalent (e.g. in fibrous proteins) and ionic (e.g. in bone mineral). Weaker hydrogen bonds are used in protein folding, making them more reversible. Since bonds are more stable at lower temperature, we could imagine organisms (perhaps in the liquid methane of Titan, or the liquid nitrogen of Triton) kept together entirely by hydrogen bonds -- effectively, living ice -- or by the even weaker Van der Waals bonds.
I already described a slime mold-like lifeform, with a body made out of isolated cells in a liquid matrix communicating through chemical tags, in my SpecBio concepts #2 and #6. May occasionally form solid structures.
Could we give up even cell membranes and have life that is truly purely liquid? We can imagine an emulsion of immiscible polar (e.g. water) and non-polar (e.g. hydrocarbons) liquids, such as we might find in pockets on a Titan-like world. The liquid with the smaller volume would form discrete chemical-concentrating droplets inside the other, perhaps switching once in a while as the relative quantities change -- living vinaigrette.
Liquid crystals are also another good way to create stable patterns without large-scale solid components. Dissolved molecules change orientation, and therefore the local properties of the fluid, in response to changes in temperature, concentration, or electric fields.
Another type of liquid (or, theoretically, gaseous) life could be based on convection cells, driven by a temperature difference (presumably geothermal at first, but the source of heat might also be exothermic reactions or radioactive decay). Bénard cells are quite regular and stable, and I can see them serving as the cells of an organism, though on a much larger scale. (There's at least one example of such life in scifi: the Qax in the Xeelee Sequence).
On the opposite end, we could imagine purely solid life, growing as extremely elaborate crystals; perhaps made of semiconductor minerals that can modulate and transport electric currents. Genetic information could be carried as patterns of doping or of imperfections in the crystalline matrix. I can imagine ecosystems of living rocks, growing mica-like photovoltaic sheets to harvest sunlight, and slowly moving by inducing magnetic fields around themselves.
It's possible that the earliest life on Earth developed on the surface of mineral layers like montmorillonite clay, using it as template for replication and as catalyzer for metabolic reactions. We could imagine life that never left this condition and developed elaborate ecosystems that are strictly two-dimensional, existing only in a one-molecule-thick film between rocks and water.
Even better: how about (partially) gaseous life? Pure gases would probably disperse chemicals too much for reactions to occur, but we could imagine living foam in which "cells" are actually high-pressure gas bubbles in a liquid medium -- or, on the contrary, living aerosol in which they are droplets of liquid suspended in a dense atmosphere, like the haze of Venus or Titan.
* Check out the links in section 3a here! Though I guess I'll have to update the links for the Astrobiology Primer 3.0 -- some asshole seems to have taken them out of free access just a couple days ago. Some interesting ideas are replacing carbon with high-temperature silicates, low-temperature silicones, nitrogen-phosphorus polymers, polysulfur, and polyoxometalates; and water with ammonia, hydrogen sulfide, sulfuric acid, low-temperature methane or nitrogen, and supercritical carbon dioxide.)
All pics from Wiki Commons. Cleander version of the atomic bonds picture from here.
