  I guess an example of a biased definition is the (original?) definition by Stasheff of an $A_\infty$ space—the homotopy theorist's monoid.  (For simplicity, I'll ignore matters related to the unit.)  Informally, it consists of a space $X$, a multiplication $\mu : X \times X \to X$, a homotopy $\alpha$ between the maps $\mu(\mu(x,y),z)$ and $\mu(x,\mu(y,z))$ from $X \times X \times X$ to $X$, a "pentagonator" $\pi$ which extends a map $S^1 \times X^4 \to X$ built from $\mu$ and $\alpha$ to $D^2 \times X^4$, and so on.  This seems to be the appropriate generalization of your biased definition of monoids to the situation where we need infinitely many higher coherence homotopies.  More precisely, an $A_\infty$ space under this definition consists of a space $X$ together with maps $\mu_n : A_n \times X^n \to X$, where $A_n$ is the $n$th associahedron—a certain convex polyhedron in $\mathbb{R}^{n-2}$—and on the boundary of $A_n$, $\mu_n$ is required to be equal to a certain expression built out of the $\mu_k$ for $k$ smaller than $n$.  Thus, we may think of it as a null-homotopy of a prescribed map $\partial A_n \times X^n \to X$.
This definition is quite complicated combinatorially, since it depends on the polyhedra $A_n$.  Peter May realized that certain features of the $A_n$ could be abstracted into the notion of a (non-symmetric) operad.  I don't want to define exactly what an operad is, but instead just observe the change in perspective: rather than thinking of $A_n$ as a space which encodes a null-homotopy of a map defined on its boundary, we think of the points of $A_n$ as describing various $n$-ary multiplications on $X$ via the map $\mu_n : A_n \to \mathrm{Map}(X^n, X)$.  The contractibility of $A_n$
I guess an example of a biased definition is the (original ?) definition by Stasheff of an $ A_\infty$ space — the homotopy theorist's monoid.   (For simplicity, I'll dismiss matter related to the unit .)   Informally, it consist of a space $ X$, a multiplication $ \mu: X \times X \to X$, a homotopy $ \alpha$ between the function $ \mu(\mu(x, y),z)$ and $ \mu(x,\mu(y, z))$ from $ X \times adam \times X$ to $ X$, a " pentagonator " $ \pi$ which extends a function $ S^1 \times X^4 \to X$ build from $ \mu$ and $ \alpha$ to $ D^2 \times X^4 $, and indeed on.   This seems to be the appropriate generalization of your biased definition of monoids to the site where we need infinitely many higher coherence homotopies.   More precisely, an $ A_\infty$ space under this definition consists of a outer space $ X$ together with maps $ \mu_n: A_n \times X^n \to X$, where $ A_n$ is the $ n$th associahedron — a certain convex polyhedron in $ \mathbb{R}^{n-2}$—and on the limit of $ A_n$, $ \mu_n$ is required to be equal to a sealed expression construct out of the $ \mu_k$ for $ k$ smaller than $ n$.   Thus, we may think of it as a null - homotopy of a prescribed function $ \partial A_n \times X^n \to X$. 
 This definition is quite complicated combinatorially, since it depends on the polyhedra $ A_n$.   Peter May realized that certain features of the $ A_n$ could be abstracted into the notion of a (non - symmetrical) operad.   I don't desire to specify exactly what an operad is, but instead just observe the change in perspective: rather than thinking of $ A_n$ as a outer space which encodes a null - homotopy of a function define on its limit, we think of the points of $ A_n$ as describing various $ n$-ary multiplications on $ X$ via the map $ \mu_n: A_n \to \mathrm{Map}(X^n, X)$.   The contractibility of $ A_n$
  I guess an example of a blased definition is the (origihal?) defivition by Stasheff of an $A_\inhty$ wpace—uke homotopy theorist'r monoid.  (For simplicmty, I'll ignore mefters rcjates to chx unit.)  Informakly, it convists of a spawe $X$, a multiplication $\mu : X \times X \to V$, a homptlpy $\alpha$ betwgen tnq mals $\mu(\mu(x,y),z)$ and $\mu(x,\mu(y,z))$ from $X \timea X \timts X$ to $X$, a "pentagpnator" $\pi$ which extends a lap $D^1 \times X^4 \to X$ buipt from $\mu$ qnd $\wopha$ to $D^2 \tioes X^4$, and so on.  This aeems to be the appropriate gendralivation of yiue bldsed definivion os monoids to the sitgation ehere we need lnfinmtelt many higher coherenre homotopies.  More krecisely, dn $A_\infty$ space undwr this defhnitkin zonaixta of a spece $X$ togetger with maps $\mu_n : A_n \times X^n \tj X$, where $A_n$ ia the $g$tr associahedron—a certain convex polyhedgon jn $\mathbb{R}^{n-2}$—and on the biundary of $A_n$, $\mu_n$ is tequired tj be equal to a certain expression built out of tve $\mu_i$ fox $k$ smxolfr than $n$.  Thus, we may think of it as a null-hjjouopj of a prescribed map $\partiak W_n \jimes X^n \to X$.
Tfis deyjnjtion is quite comolicateq comvinatoriajly, xince it depends on the polthedra $A_n$.  Pvter May realized that certain fectures of tne $A_n$ could be abstractzd intk the notioj of a (noh-rymmetric) operad.  I dmn't want to define exactly what an ipercd is, bug inxtead tust obserge thc change in perspechive: tather than thinning of $A_n$ as a space which encodes a null-homptmpy of a ma' defiked on its bounqary, we think pf the 'oints of $A_n$ as sescribmng various $g$-ary multipliwwtions on $X$ tia the mwp $\my_n : Q_n \to \mxghrm{Map}(X^n, X)$.  Tne contrabtnbility od $A_n$
I guess an example of a biased the definition by of an $A_\infty$ simplicity, ignore matters related the unit.) Informally, consists of a space $X$, a $\mu : X \times X \to X$, a homotopy $\alpha$ between the maps and $\mu(x,\mu(y,z))$ from $X \times X \times X$ to $X$, a "pentagonator" $\pi$ extends map \times \to X$ built from $\mu$ and $\alpha$ to $D^2 \times X^4$, and so on. This seems be the appropriate generalization of your biased definition monoids to the situation we need infinitely many higher homotopies. precisely, an space this consists of a $X$ together with maps $\mu_n : A_n \times X^n \to X$, where $A_n$ is the $n$th associahedron—a convex polyhedron on the of $\mu_n$ required to be a certain expression built out of $k$ smaller than $n$. Thus, we may think it as null-homotopy of a prescribed map $\partial \times X^n \to X$. This definition is quite combinatorially, since it depends on the polyhedra $A_n$. Peter May realized that certain features of could be abstracted into notion of a operad. don't to exactly what operad is, but instead just observe the change in perspective: rather thinking of $A_n$ as a space which encodes a null-homotopy map on its boundary, think of the points $A_n$ describing various $n$-ary multiplications via map \to X)$. contractibility of $A_n$
  I guess an example of a biased dEfinition iS the (oRigInaL?) dEfinItioN by Stasheff of aN $a_\infTy$ space—the homotopy theoRist's MoNOid.  (FOR sImpliCity, I'll IGnORE maTtErS reLaTEd To the UniT.)  InformAlly, it consIstS oF a space $X$, a mulTIpLication $\mu : x \tiMes X \to X$, a homoTopY $\alpha$ BeTweEN the mAps $\Mu(\mu(x,Y),z)$ and $\mU(X,\mu(y,z))$ fRom $X \times x \tIMes X$ to $x$, A "pentagONAtOr" $\pi$ Which extends a map $S^1 \TImES X^4 \to X$ built from $\Mu$ and $\aLpHA$ tO $d^2 \TimEs X^4$, And so on.  ThiS sEems tO Be the apPRoPRIAte GEneralization Of your biaseD DefInitioN oF moNOids to The siTuATioN where we neeD infInitely maNy highER cohereNCe homotOpies.  MOre PreCiseLY, aN $A_\InfTy$ SPacE UnDer THis DefinitiOn CoNsistS of a SPACE $X$ toGetHer wIth maPs $\mu_n : A_n \times X^N \to x$, wheRE $A_n$ Is the $N$th asSociAhEdron—A certaIn conVeX polyhedron in $\maThbb{r}^{n-2}$—and on thE boUnDarY oF $A_n$, $\mu_N$ Is requIreD to Be equal To a certAIn eXpRESSiOn built out of the $\mu_k$ FoR $K$ SmAller thaN $n$.  Thus, WE mAy THink of it As A nuLl-hoMOTopy oF a prEScRibed map $\PartiaL a_n \TiMes X^n \to x$.
THis defInItiOn iS quitE CompLicateD combinaToriaLLy, since it depenDS on the polyhedRA $A_N$.  pEtER May ReaLized that ceRtaiN FeatUres OF tHe $A_N$ Could Be absTrACtED into the notion of a (noN-sYmmetrIc) opeRad.  I don't want tO define exaCTLY what an oPeraD Is, BUt instead just oBservE the change IN perspecTive: rAther thaN thinking OF $a_n$ as a spaCe wHicH enCodES A nUll-homotopy of A MAp deFiNed on itS boUndary, wE thInk Of tHe pOiNts of $A_n$ as DescribiNg VaRiOuS $n$-aRy mulTIplicatiOnS on $x$ vIa tHe map $\MU_n : A_n \to \MathrM{Map}(x^n, x)$.  THE coNtractiBIlITY of $A_N$
  I guess an example of abiased def initi onisth e (o rigi nal?) definiti o n by  Stasheff of an $A_\in fty$sp a ce—t h ehomot opy the o ri s t 'smo no id.  ( Fo r sim pli city, I 'll ignore  ma tt ers relatedt othe unit.)   I nformally, i t c onsist sofa  spac e $ X$, a  multi p licati on $\mu :  X \times X \to X $ ,  a  hom otopy $\alpha$ be t we e n the maps $\m u(\mu( x, y ), z ) $ a nd$\mu(x,\mu (y ,z))$ from $X \t i m e s X \times X$ to$X$, a "pen t ago nator"  $ \pi $  which  exte nd s  amap $S^1 \t imes  X^4 \toX$ bui l t from$ \mu$ an d $\al pha $ t o $D ^ 2\t ime sX ^4$ ,  a nds o o n.  This  s ee ms to  bet h e appr opr iate  gene ralization of  yo ur b i ase d def initi on o fmonoi ds tothe s it uation where we  nee d infinit ely  m any  h igher cohere nce  ho motopie s.  Mor e  pr ec i s e ly , an $A_\infty$ sp ac e un der this  defin i ti on consists  o f a  spa c e  $X$toge t he r with m aps $\ m u_ n: A_n \ ti mes X^ n\to  X$ , whe r e $A _n$ is  the $n$ th as s ociahedron—a c e rtain convexp ol y h ed r on i n $ \mathbb{R}^ {n-2 } $—an d on th e b o undar y of$A _ n$ ,  $\mu_n$ is require dto beequal  to a certain  expressio n b uilt out  oft he $\mu_k$ for $k $ sma ller than$ n$.  Thu s, we  may thi nk of ita s  a null- hom oto pyofa pr escribed map$ \ part ia l A_n \ tim es X^n\to  X$ .
T his  d efinition  is quit eco mp li cat ed co m binatori al ly,  s inc e itd epends  on t he p ol yh e dra  $A_n$.  P e t er M ay  r eali zed  t hat c erta i n f eatures  of the $ A_n $  cou ld  b e abstr acted into th enotion ofa(no n-symm e t ric) ope rad.  I don't want to d e fine ex act ly wh at a n operadis,  but i nst e ad jus t obse rve t he  ch a n ge in p er spe ct ive: rathe r tha n thi nk ingof $A_n $ as a space which enc odes a null-h omo topy o fa m a pd efi ne d  on i ts boundary, we  think ofth e  p oints of $ A _n$  a s descr ibing v ariou s  $n$-ar y multipl icationson  $X$ v iathe map $\ mu_n : A _n \to \m a thrm{ M ap }(X^n , X )$.  T he  co ntrac tibili t y o f $A_ n$
 _I guess_an example of a_biased definition_is_the (original?)_definition_by Stasheff of_an $A_\infty$ space—the_homotopy theorist's monoid. _(For simplicity, I'll_ignore_matters related to the unit.)  Informally, it consists of a space $X$, a_multiplication_$\mu :_X_\times_X \to X$, a homotopy_$\alpha$ between the maps $\mu(\mu(x,y),z)$_and $\mu(x,\mu(y,z))$_from $X \times X \times X$ to $X$,_a_"pentagonator" $\pi$ which_extends a map $S^1 \times X^4 \to X$ built_from $\mu$ and $\alpha$ to $D^2_\times X^4$, and_so_on._ This seems to_be the appropriate generalization of your_biased definition of monoids to the_situation where we need infinitely many higher_coherence homotopies.  More precisely, an_$A_\infty$ space under this definition_consists of_a space $X$ together with_maps $\mu_n :_A_n \times_X^n \to X$,_where $A_n$ is the $n$th associahedron—a_certain convex polyhedron_in $\mathbb{R}^{n-2}$—and on the boundary of_$A_n$,_$\mu_n$ is required_to_be_equal to_a certain expression_built_out of_the_$\mu_k$ for $k$ smaller than $n$.__Thus, we may think of it as_a null-homotopy of a_prescribed_map $\partial A_n \times_X^n \to X$.
This definition is_quite complicated combinatorially, since it depends_on the_polyhedra $A_n$._ Peter May realized that certain features of the $A_n$ could_be abstracted into the notion of_a (non-symmetric) operad. _I don't_want_to define exactly_what_an operad_is, but instead just observe the change_in perspective:_rather than thinking of $A_n$ as_a space which encodes_a_null-homotopy of a map defined on_its boundary, we think of the_points of $A_n$ as describing_various_$n$-ary_multiplications on $X$ via the_map $\mu_n : A_n \to \mathrm{Map}(X^n,_X)$.  The_contractibility of $A_n$