Hyperoperations can be extended to transfinite ordinals
Let's define
1) ,
2) ,
3) iff is a limit ordinal.
Or in such form
1) ,
2) ,
3) iff is a limit ordinal,
where denotes the bth element of the fundamental sequence assigned to the limit ordinal .
 Third rule was inspired by Aeton's work.
Examples
and for arbitrary power of omega
,
.
Nomenclature
Abbreviations adopted for the building of names:
add=addition, ult=multiplication, ex=exponentiation, etr=tetration, om=omega, phi  binary Veblen function
un,b,tr,quadr,quint,sext,sept,oct,non,dek=1,2,3,4,5,6,7,8,9,10
Although names of those hyper operators are very similar to names of my numbers, do not confuse them  just for the creation of names of hyperoperations almost same abbreviations were used as in my nomenclature of numbers.
unaddomation
a unaddomated to b  
baddomation
(a baddomated to b)  bultomation
(a bultomated to b)  bexomation
(a bexomated to b)  betromation
(a betromated to b) 
traddomation
(a traddomated to b)  trultomation
(a trultomated to b)  trexomation
(a trexomated to b)  tretromation
(a tretromated to b) 
quadraddomation
(a quadraddomated to b)  quadrultomation
(a quadrultomated to b)  quadrexomation
(a quadrexomated to b)  quadretromation
(a quadretromated to b) 
quintaddomation
(a quintaddomated to b)  quintultomation
(a quintultomated to b)  quintexomation
(a quintexomated to b)  quintetromation
(a quintetromated to b) 
sextaddomation
(a sextaddomated to b)  sextultomation
(a sextultomated to b)  sextexomation
(a sextexomated to b)  sextetromation
(a sextetromated to b) 
septaddomation
(a septaddomated to b)  septultomation
(a septultomated to b)  septexomation
(a septexomated to b)  septetromation
(a septetromated to b) 
octaddomation
(a octaddomated to b)  octultomation
(a octultomated to b)  octexomation
(a octexomated to b)  octetromation
(a octetromated to b) 
nonaddomation
(a nonaddomated to b)  nonultomation
(a nonultomated to b)  nonexomation
(a nonexomated to b)  nonetromation
(a nonetromated to b) 
dekaddomation
(a dekaddomated to b)  dekultomation
(a dekultomated to b)  dekexomation
(a dekexomated to b)  deketromation
(a deketromated to b) 
uniphiation  unomtation 
biphiation  bomtation 
triphiation  tromtation 
quadriphiation  quadromtation 
quintiphiation  quintomtation 
sextiphiation  sextomtation 
septiphiation  septomtation 
octiphiation  octomtation 
noniphiation  nonomtation 
dekophiation  dekomtation 
Note: is the Veblen function and is abbreviation for Feferman theta function .
Example for using inaccessible cardinal: let's calculate for n=1,2 (probably this operation should be named ipsitation). Remarkable that for this values of n we will obtain modest output.
Here is abriviation for and
, where is function such that and for ,
also note that ,


Post Scriptum: Also let's introduce function .