How can I remove autocorrelation from this data? I have 1000 data points and can see there is quite a high level of autocorrelation. 
If I look at the lag 1 data pairs (1, 2), (3, 4), (5, 6), ..., then the correlation is r = .62. 
I put the raw data here. I'd like to transform the data to remove or reduce the autocorrelation. 
The context for all this is that the data points are guesses made by individuals about some quantity. I know the true value of the quantity and want to see whether the average guess is better if I just leave the data autocorrelated, or if I remove the autocorrelation.
One idea I had for an approach was to get rid of the lag 1 autocorrelation first, and then to see if average accuracy improves, and then try to get rid of the lag 2 autocorrelation.
I thought maybe I could just throw out every second data point, and then naturally autocorrelation would reduce. However, ideally I'd like to find a way to remove autocorrelation without throwing away data.
 A: The setting looks like a regression with ARMA errors, more precisely AR(1) errors (by construction, as I understand the data generating process from your comment). It can be estimated by function arima in R specifying ARIMA order (1,0,0) and including any regressors as exogenous regressors via the argument xreg. The adjusted data could be obtained as residuals from the model using residuals(model) where model is the estimated model, plus the intercept of the model (since in arima intercept is actually the mean, see here, ISSUE 1).
Your approach of deleting data point is not a very good idea. You will lose a lot of information but the autocorrelation will still be there.
A: dta <- read.table("anchoring.csv", col.names="v1")
summary(lm1 <- lm(dta$v1[-1]~dta$v1[-1000]))

This is an AR(1) regression, estimated using lm instead of arima. The summary shows that ~40% of the variance in the persons' guesses can be explained by the guesses they were shown (that is the R squared). 
predict(lm1)

will give you the expected value of a subject's guess, given the guess shown to them.
