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I faced with issue that sometimes when I created custom formula-fields on Salesforce objects I couldn't save it, the reason of it was limitation in 5000 characters for such type of field.

The main trouble is that when I copied content of formula to any notepad which can calculate number of characters I saw, that there was less than 5000 characters. After some investigations I found that mentions of other formula-fields and also some methods, like TODAY(), can implicitly increase number of characters. In this way the real length will be more than number of character that you type.

My question - how can I see the real amount of formula-field characters and how to know which parts of formula adds extra-amount?

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I believe your are talking about the compile size of 5000 bytes

Compile size limit—Formula fields can’t exceed 5,000 bytes when compiled. The compile size is the size of the formula (in bytes) including all of the fields, values, and formulas it references. There is no direct correlation between the compile size and the character limit. Some functions, such as TEXT, DATEVALUE, DATETIMEVALUE, and DATE significantly increase the compile size.

Note the part about no direct correlation between number of characters and the compiled bytes.

and not the character limit which is 3900 characters per formula.

Character limit—Formula fields can contain up to 3,900 characters, including spaces, return characters, and comments. If your formula needs more characters, create separate formula fields and reference them in another formula field.

Which can be found in the DOCS:

https://help.salesforce.com/HTViewHelpDoc?id=formula_field_limits.htm&language=en_US

Additional information from a simple search here as well:

https://salesforce.stackexchange.com/search?q=formula+compile+size

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