Is there any real number that starts with J?

No - at least not in the English language.

We have

zero (starts with z)

one (starts with o)

two (starts with t)

three (starts with t)

four (starts with f)

five (starts with f)

six (starts with s)

seven (starts with s)

eight (starts with e)

nine (starts with n)

ten (starts with t)

eleven (starts with e)

twelve (starts with t)

thirteen (starts with t)

fourteen (starts with f)

fifteen (starts with f)

sixteen (starts with s)

seventeen (starts with s)

eighteen (starts with e)

nineteen (starts with n)

twenty (starts with t)

thirty (starts with t)

forty (starts with f)

fifty (starts with f)

sixty (starts with s)

seventy (starts with s)

eighty (starts with e)

ninety (starts with n)

from there on up, all numbers start with one of these, for example:

35,500,400,122

Thirty five billion five hundred million, four hundred thousand one hundred twenty-two (starts with T)

We do have some number related Words that we don't actually start the names of numbers with:

Hundred

Thousand

Million

Billion

Trillion

Quadrillion

Quintillion

Sextillion

Septillion

Octillion

Nonillion

Decillion

Undecillion

Duodecillion

Tredecillion

Quatrodecillion

Virgintillion

...

and so forth continuing to use the same types of prefixes with u (un), b(bi) d(duo,dec), t(tri), q(quad, quint), s(sex, sept), o(oct), n(non), c(cent), m(mili), None of them start with "J".

The extremely large numbers Googol and googleplex both start with G.

The metric prefixes include

kilo

mega

giga

tera

peta

exa

zeta

yotta

(none of which start with J)

and fractional prefixes of

deci

centi

milli

micro

nano

pico

femto

atto

zepto

yocto

(still nothing starting with J)

Informally 10 to the 27th power is called "hella" - as sort of a nerd joke - but still no "J" number.