The Complication Most Worth Testing For ...

is retinopathy (in diabetics of pubertal age and beyond).

There's little to no evidence that screening for kidney disease or neuropathy changes what happens with that kidney disease or neuropathy (although theoretically, once you know you have neuropathy, you should be taking more precautions). The things you can do for early, asymptomatic, kidney disease can be done without any knowledge that you have kidney disease (mostly, manage blood pressure), and there's nothing to do for asymptomatic neuropathy.

However, asymptomatic bleeding in the eyes can be treated.

The study that the ADA quotes in its 2013 statement about the worth of treating diabetic retinopathy with panretinal photocoagulation, saying that it reduces the risk of blindness in that eye from about 16% to about 6%, is an interesting study.
The Diabetic Retinopathy Study began in 1971, and it enrolled 1742 adults under the age of 70 with severe nonproliferative retinopathy and/or proliferative retinopathy, in both eyes. They randomly had one eye get treated, and one eye not get treated, which is a pretty cool control group IMHO.
The authors of this study believe that treating diabetic retinopathy should be best done neither early nor late; that it should be treated before vision loss occurs, but not in early stages of retinopathy, because of the risk of the treatments.
The value of knowing that a person has mild retinopathy is mostly only in being able to test for more severe retinopathy more often; retinopathy rarely develops very quickly.

There is some controversy/ discussion in medical journals about how often screening for retinopathy should be done, particularly in diabetics without retinopathy. Although the current clinic recommendation most often given is once per year, I think the evidence is pretty strong that screening every other year is sufficient... and I saw an eye doctor recently who thinks type 1 diabetics should be screened twice per year!

A metastudy: http://www.ncbi.nlm.nih.gov/pubmed/23819487
Study listing with links: http://clinicaltrials.gov/show/NCT00000160