Could you show an example of that, please?

An example of what? Of using such types?
I use them in numerical methods of solution of an initial value problem. In some cases the usual "double" type is not precise enough.
Also I use interval arithmetics to control rounding errors.
I have an idea to store calculation data (for example, time step) in a database. I know only one opensource DBMS which supports a numeric type with a long mantissa: PostgreSQL (NUMERIC type http://www.postgresql.org/docs/9.3/static/datatype-numeric.html).

ah, ok. This is our DECIMAL() data type, it already exists. See https://mariadb.com/kb/en/decimal

If I am not mistaken, it is a fixed-point type and the maximum number of digits is 65. So this type cannot represent such numbers as 10^300.
I need a floating point type (like "double" but with user selected mantissa length).

NUMERIC type in PostgreSQL is a fixed-point type too, as far as I can see. Our best match is DECIMAL type. But it doesn't support 300 digits of precision, indeed. So I suppose you want longer DECIMAL numbers, right?

No, I want a type like "REAL" but with a user selected precision (mantissa length). An exponenta length can be user selected or unlimited. For example, Boost (http://www.boost.org/doc/libs/1_55_0/libs/multiprecision/doc/html/boost_multiprecision/tut/floats.html) and GNU MPFR deals with such numbers.