The BEA describes the differences between its measures of government spending, noting that "Total spending by government is much larger than the spending included in GDP. " with its "total expenditures" figures including the most.
Most figures on government spending are incomplete
Unfortunately there don't appear to be complete figures for the true total of government spending to get an idea of what fraction of the economy it truly represents. e.g. the federal BEA provides a figure for "Total expenditures of government" (federal, state, and local combined) of $5,642.9 for 2011 using their usual NIPA (National Income and Product Accounts) accounting approach. However they also report data using the international SNA (System of National Accounts) approach which gives $6,251.4 billion in total expenditures. That is higher by about 11%, $608.5 billion, and the descriptions of its process indicate that is still too low since they hide some spending in non-government categories The portion of government funded by taxes appears to all be included, which should be sufficient for forecasts of future deficits and taxes vs. spending.
The largest issue is that when receives any sort of fees for services, it often uses that as an excuse to hide expenses. The federal BEA (Bureau of Economic Analysis) provides a "A Primer on GDP and the National Income and Product Accounts" where they describe the components of national accounts. As it mentions, they hide many government activities in the business sector: "Businesses [... ]The sector comprises [...] and government enterprises. Government enterprises are government agencies—such as the U.S. Postal Service or state government-run utilities—that cover a substantial portion of their operating costs by selling goods and services to the public." In their more detailed description of government accounts they note: "Government enterprises are government institutional units that are classified in the business sector because they sell their goods and services directly to the public for an economically significant price". i.e. if they nationalized an industry, most of their accounting figures would still classify it as a "business" even if it was under political control and heavily subsidized to benefit special interest groups. If all private companies were taken over by a government, most figures would still pretend they were in the "business" sector.
They note a related game government plays to hide expenses when they describe differences between their figures and the federal budget: "First, the budget nets certain receipts against outlays whereas the NIPAs record a receipt". e.g. if the feds have a fee of $10 for a service that costs $20 they might only record a $10 expense rather than $20 in expenses and $10 in receipts. The FY2013 proposed federal budget has a Chapter 16 on "Offsetting Collections and Offsetting Receipts" where it notes: "For 2011 gross outlays to the pubic were $4152 billion [...] net outlays were $3,603 [...] Offsetting collections and offsetting receipts from the public are subtracted from gross outlays to the public to yield “net outlays,” which is the most common measure of outlays cited and generally referred to as simply 'outlays' ". The actual proposed budget figures are "net outlays" and the budget's "Historical tables" page has categories it claims are "Total outlays" or "Total Government Expenditures" but which are actually only "net outlays". There doesn't seem to be a collection of historical "gross outlay" data.
The BEA plays the same game, merely in different places than the budget. They note "Second, in some cases the budget records a receipt while the NIPAs record an offset against current expenditures.". There are other difference in their methodology which impact a direct comparison, but for 2011 they only give $3,923.2 for "total expenditures" for the federal government. The BEA data using the international SNA standard isn't much different at $3,940.2 billion for 2011 for "central government" total expenditures. The major difference between the NIPA and SNA figures for "total government" expenditures arise from state&local spending.
The BEA republishes Census Bureau figures for state&local government expenditures which are rather different from its own figures. The lastest comparison they have in Table 3.19 is for 2010 when the Census Bureau gives a figure of $3,114.8 billion while the BEA's figure is $2,112.6 billion for "current expenditures" (elsewhere "total expenditures" for state&local are $2,278.6 billion). Using the international SNA approach they report $2,842.2 billion for 2010. When describing the differences it notes the same issue with hiding expenditures of government enterprises: "In the NIPAs, expenditures are generally shown net of related sales revenue" among other issues like: "These differences include transactions associated with state and local government retirement plans, which the NIPAs account for in the household, not government sector". Another difference it gives is that unemployment insurance is treated as a state expense in the Census figures and a federal expense in the BEA figures.
In a description of the SNA on the UN site they give a more concrete criteria for government enterprises than the NIPA has: "9.34. As long as those entities can charge market prices or prices that cover over 50 percent of costs, they are excluded from the government sector." i.e. a government could nationalize an industry, subsidize 49% of it, and it would still be kept in the "business sector". International "total expenditure" figures don't even in theory give an idea of how much of other country's economies are under government control. Obviously even for figures that are reported, its possible some countries may try to find ways to hide how much they do spend.
It would take more research to determine whether there is enough information available from the BEA and Census Bureau or other sources to get a better figure for the fraction of the economy under government control.