Aside from Redshift, what is the most compelling evidence that the universe is expanding?

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Airtime:

'We are the stuff of the universe, momentarily sentient, but aside from that little piece of weirdness,we are already home.'
Jennifer Michael Hecht

The Tolman dimming and, less directly, the cosmic microwave background. On top of these, the fact that the volume densities of assorted objects increase with redshift at least as fast as the (1+z)^3 expected from cosmic expansion (much more rapidly for things which show evidence of strong evolution; the more boring the galaxy type, the closer it comes to this). And related to the Tolman test, the fact that supernova absolute magnitudes are much less constant with redshift if you leave out the photon-rate and energy-shift terms that come from redshift in an expanding spacetime (the former, as far as I know, uniquely such a prediction).

I think you ask the wrong question. It is not usually the case that you can pick one "evidence" in any physical argument and call it convincing, by itself. Rather, it is the whole collection of evidence, all consistent with the one idea that really gives the one idea strength.

Before the redshift-distance relationship was discovered by Edwin Hubble (Hubble, 1929), the expansion of the universe had already been predicted by the Belgian cleric & astronomer Georges Lemaitre, based on the implications of the theory of general relativity (Lemaitre, 1931a, Lemaitre 1931b, Lemaitre 1934; his original 1927 paper is not available online). The interpretation of the observed redshift distance relationship as an expansion of the universe is fairly obvious, even without any precedent in theory. But the combination of theory and observation strengthens the interpretation of expansion.

Since then we have learned to understand far more about general relativity than Einstein & his contemporaries understood (i.e., Will, 2006) but continue to learn more (i.e., Davis & Lineweaver, 2004). In 1948 George Gamow and Ralph Alpher combined general relativity & the expanding universe to demonstrate how this physical scenario could reproduce the relative abundance of hydrogen & helium in the universe, and to predict that the present universe would be filled with a background electromagnetic radiation left over from the infancy of the universe (Alpher, Bethe & Gamow, 1948; Genesis of the Big Bang by Ralph Alpher & Robert Herman, Oxford University Press, 2001). The relative abundances have blossomed into the science of Big Bang Nucleosynthesis (BBN) and we now recognize the ubiquitous background as the Cosmic Microwave Background (CMB).

So we have an observed redshift distance relationship, and observed CMB, and observed relative abundances of elements, and the theory of general relativity, all of which are mutually consistent with the idea of an expanding universe. It is this mutual consistency between independent lines of evidence that makes Big Bang Cosmology a strong idea, and not necessarily any one of those lines of evidence considered by itself.

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Don't try this at home - We're what you call "professionals" - MythBusters.

that makes good sense (thanks)

One aspect of the expanding universe concept that I have puzzled over each time I see it expressed, is the idea that if the universe was not expanding, then it would collapse in on itself.

If the universe was a finite bounded entity then that idea makes sense, but if the universe was infinite and unbounded, then the concept of collapse appears to me to have no meaning.
Sure, parts where there are concentrations of matter could collapse together in clumps (whereupon they would eventually blow apart or form black holes I thought), but how can something truly infinite collapse?

It is usually expressed as a self-obvious preamble to the deeper thoughts that lead to relativity etc and something that had occurred to numerous people at the time, but for an infinite, non-expanding universe as it was thought to be at that time, the idea seems to make no sense.

__________________
Airtime:

'We are the stuff of the universe, momentarily sentient, but aside from that little piece of weirdness,we are already home.'
Jennifer Michael Hecht

My waistline.

And someday, we'll try to measure that "expansion" and its variations directly, and that would be really cool.