The following illustration should help:
An old thread, but a useful one... so here's some additional code.
First, in response to Craig's point about the regex being overly greedy and failing for sheet names containing a single quote, this should do the trick (replace 'SHEETNAME'!A1 with your own sheet & cell reference):
=IF(TODAY()=TODAY(), SUBSTITUTE(REGEXREPLACE(CELL("address",'SHEETNAME'!A1),"'?(.+?)'?!\$.*","$1"),"''","'", ""), "")
It uses a lazy match (the ".+?") to find a character string (squotes included) that may or may not be enclosed by squotes but is definitely terminated by bang dollar ("!$") followed by any number of characters. Google Sheets actually protects squotes within a sheet name by appending another squote (as in ''), so the SUBSTITUTE is needed to reduce these back to single squotes.
The formula also allows for sheet names that contain bangs ("!"), but will fail for names using bang dollars ("!$") - if you really need to make your sheet names to look like full absolute cell references then put a separating character between the bang and the dollar (such as a space).
Note that it will only work correctly when pointed at a different sheet from the one that the formula resides! This is because CELL("address" returns just the cell reference (not the sheet name) when used on the same sheet. If you need a sheet to show its own name then put the formula in a cell on another sheet, point it at your target sheet, and then reference the formula cell from the target sheet. I often have a "Meta" sheet in my workbooks to hold settings, common values, database matching criteria, etc so that's also where I put this formula.
As others have said many times above, Google Sheets will only notice changes to the sheet name if you set the workbook's recalculation to "On change and every minute" which you can find on the File|Settings|Calculation menu. It can take up to a whole minute for the change to be picked up.
Secondly, if like me you happen to need an inter-operable formula that works on both Google Sheets and Excel (which for older versions at least doesn't have the REGEXREPLACE function), try:
=IF(IFERROR(INFO("release"), 0)=0, IF(TODAY()=TODAY(), SUBSTITUTE(REGEXREPLACE(CELL("address",'SHEETNAME'!A1),"'?(.+?)'?!\$.*","$1"),"''","'", ""), ""), MID(CELL("filename",'SHEETNAME'!A1),FIND("]",CELL("filename",'SHEETNAME'!A1))+1,255))
This uses INFO("release") to determine which platform we are on... Excel returns a number >0 whereas Google Sheets does not implement the INFO function and generates an error which the formula traps into a 0 and uses for numerical comparison. The Google code branch is as above.
For clarity and completeness, this is the Excel-only version (which does correctly return the name of the sheet it resides on):
=MID(CELL("filename",'SHEETNAME'!A1),FIND("]",CELL("filename",'SHEETNAME'!A1))+1,255)
It looks for the "]" filename terminator in the output of CELL("filename" and extracts the sheet name from the remaining part of the string using the MID function. Excel doesn't allow sheet names to contain "]" so this works for all possible sheet names. In the inter-operable version, Excel is happy to be fed a call to the non-existent REGEXREPLACE function because it never gets to execute the Google code branch.