How should I test for autocorrelation in this time series context? I have data sets in which different people estimate a certain quantity. They potentially can see the estimates of anyone who participated before them, but in practice they're only likely to look at the last few estimates that were made before them. For that reason I am expecting some autocorrelation.
Here's an example in which 199 people (x-axis) made estimates mostly falling between 900 and 1300. There is little visual evidence of autocorrelation here, but autocorrelation does seem to be present in some of my other data sets.

I'm interested to test for the level of autocorrelation and if it is high to examine the effects of reducing it, either by providing different instructions to the participants or by performing some statistical manipulation to the data. 
I'm familiar with testing for autocorrelation using the Durbin-Watson test in a regression context, but am not sure how to proceed here. Almost all google searches related to autocorrelation seem to relate to a regression context.
 A: You could assume that if there is any autocorrelation at all, there will definitely be autocorrelation at lag 1. The null hypothesis to test would be that the data are white noise, and the alternative is that there is autocorrelation including $\rho(1) \neq 0$.
Therefore you can estimate the correlation at lag 1 and apply the standard t-based test for correlation. In the estimation you can only use non-overlapping pairs of samples $(1,2)$, $(3,4)$, … so that sample pairs are independent under H0. If you are worried about nonnormality, use Spearman correlation and use an appropriate test for that.
To avoid having to use non-overlapping pairs of samples, you could also determine the null distribution empirically by comparing the actual correlation to the distribution that arises in timeseries that are generated by resampling with replacement (testing the null hypothesis that the time series is stationary white noise).

Here are concrete results based on the actual data provided in a comment, analyzed using Matlab's corr and xcov. The timeseries of 197 samples is stored in the vector x.
lag-1 autocorrelation: to compute without overlap, so that the sample pairs are independent under the null hypothesis of white noise:
>> [rho, pval] = corr(x(1 :2: end - 1), x(2 :2: end))
rho =
       -0.0262418232219902
pval =
         0.797571094920343

Looking at the data we find that there are three outliers that may influence the test implemented in corr, which is designed for normally distributed data. To double check, we use the test based on the Spearman rank correlation:
>> [rho, pval] = corr(x(1 :2: end - 1), x(2 :2: end), 'type', 'Spearman')
rho =
       -0.0431684721353655
pval =
          0.67298135968005

The result is very similar to that based on Pearson correlation. We therefore have no evidence that there is autocorrelation at lag 1.
autocorrelation function: But maybe our assumption – that there if there is any autocorrelation at all, this will include lag 1 – is not correct. Let's look at the autocorrelation function:
[r, lags] = xcov(x, 'coeff');

figure
plot(lags, r, 'b.-')
ylim([-0.3 1.05])
xlabel lags
ylabel correlation


Now it looks like the autocorrelation at lag 1 is exceptionally small. The strongest autocorrelation is -0.215 at lag 21. Of course we cannot be sure that this is not just a randomly (negative-) large value. To test whether the autocorrelation at some other lag is significantly different from zero, we simulate stationary white noise with the same distribution as the data by resampling with replacement, and estimate the probability to find an autocorrelation larger than 0.215 or smaller than -0.215 at any non-zero lag:
[r, lags] = xcov(x, 'coeff');
mar = max(abs(r(lags > 0)))

for i = 1 : 100000
    xb = x(randi(n, n, 1));
    [rb, lags] = xcov(xb, 'coeff');
    marb(i) = max(abs(rb(lags > 0)));
end

pval = mean(marb >= mar)

The result is a p-value of 0.11 – much smaller, but still not significant w.r.t. the conventional level 0.05. From the distribution of simulated maximum-absolute-nonzero-lag-correlations marb we can determine the critical value of a test at a level of 0.05
>> rc = quantile(marb, 0.95)
rc =
         0.233023218494253

and visualize it

To conclude, according to this analysis, your data do not show a statistically significant amount of autocorrelation.
